Multiplicity Results of Solutions to the Fractional p-Laplacian Problems of the Kirchhoff–Schrödinger–Hardy Type

Abstract

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively.

1. Introduction and Main Results

The present paper is dedicated to demonstrating multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials:

$$-M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right){\mathcal{L}}_p^s\varphi \left(x\right)+\mathfrak{a}\left(x\right){|\varphi |}^{p-2}\varphi =\lambda {\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}+g(x,\varphi )\mathrm{in}{ℝ}^N,$$

(1)

where $0<s<1<p<+\infty$ with $sp<N$, ${\left[\varphi \right]}_{p,\mathcal{K}}:={\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz$, $M\in C\left({ℝ}^{+}\right)$ is a Kirchhoff type function, $\lambda$ is a real parameter, a potential function $\mathfrak{a}:{ℝ}^N\to (0,\infty )$ satisfies

(V)

$\mathfrak{a}\in C\left({ℝ}^N\right)$, ${ess\; inf}_{x\in {ℝ}^N}\mathfrak{a}\left(x\right)>0$, and $\mathrm{meas}\left\{x\in {ℝ}^N:\mathfrak{a}\left(x\right)\le {\mathcal{C}}_0\right\}<+\infty$ for any ${\mathcal{C}}_0>0$.

• and a Carathéodory function $g:{ℝ}^N\times ℝ\to ℝ$ has a subcritical nonlinearity. Here nonlocal operator ${\mathcal{L}}_p^s$ is defined pointwise as

$${\mathcal{L}}_p^s\varphi \left(x\right)=2{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))\mathcal{K}(x,z)dz\mathrm{for}\mathrm{all}x\in {ℝ}^N,$$

where a kernel function $\mathcal{K}:{ℝ}^N\times {ℝ}^N\to (0,+\infty )$ admits the following properties:

(L1)

$\kappa \mathcal{K}\in L^1({ℝ}^N\times {ℝ}^N)$, where $\kappa (x,z)=min\{|x-z{|}^p,1\}$;

(L2)

there is ${\gamma }_0>0$ such that ${\mathcal{K}(x,z)|x-z|}^{N+sp}\ge {\gamma }_0$ for almost all $(x,z)\in {ℝ}^N\times {ℝ}^N$ with $x\ne z$;

(L3)

$\mathcal{K}(x,z)=\mathcal{K}(z,x)$ for all $(x,z)\in {ℝ}^N\times {ℝ}^N$.

Also, we suppose that g satisfies the hypotheses as follows:

(Ψ1)

$g:{ℝ}^N\times ℝ\to ℝ$ satisfies the Carathéodory condition and there exist ${\sigma }_1>0$, $r\in (p,p_s^{*})$, and $0\le {\sigma }_0\in L^{r^{\prime }}\left({ℝ}^N\right)\cap L^{\infty }\left({ℝ}^N\right)$ such that

$$|g(x,\zeta )|\le {\sigma }_0\left(x\right)+{\sigma }_1{|\zeta |}^{q-1}$$

for all $(x,\zeta )\in {ℝ}^N\times ℝ$, where $q\in (p,p_s^{*})$.

(Ψ2)

${lim}_{|\zeta |\to \infty }{\displaystyle \frac{G(x,\zeta )}{{|\zeta |}^{\vartheta p}}}=\infty$ uniformly for almost all $x\in {ℝ}^N$.

(Ψ3)

$g(x,-\zeta )=-g(x,\zeta )$ holds for all $(x,\zeta )\in {ℝ}^N\times ℝ$.

(Ψ4)

There are constants $C>0$, $\kappa >1$, $\tau >1$ with $\kappa <p\le {\tau }^{\prime }\kappa \le p_s^{*}$, and $0<\eta \in L^{\tau }\left({ℝ}^N\right)\cap L^{\infty }\left({ℝ}^N\right)$ such that

$$\underset{|\zeta |\to 0}{lim\; inf}{\displaystyle \frac{g(x,\zeta )}{{\eta \left(x\right)|\zeta |}^{\kappa -2}\zeta }}\ge C$$

uniformly for almost all $x\in {ℝ}^N$.

In the last few decades, considerable attention has been devoted to investigating problems associated with fractional Sobolev spaces and the corresponding nonlocal equations because this type of problem arises in various fields of mathematical physics, such as image reconstruction, phase transition models, finance, optimization, anomalous diffusion, stratified materials, and crystal dislocation, among others. For more details, refer to [1,2,3,4,5,6,7] and references therein.

One interesting aspect of our problem is the appearance of Hardy potentials. In recent years, stationary problems involving singular coefficients have gained an increasing amount of attention because they can be corroborated as a model for several physical phenomena and applied economical models; see [8,9,10] for more comprehensive details and examples. In addition, thanks to this great interest, such problems have been studied more in recent years; see [11,12,13,14].

From a mathematical point of view, these elliptic problems involving singular coefficients pose some technical difficulties since this operator is not homogeneous and the sequences of the Palais–Smale type do not ensure the compactness property. In particular, showing the compactness condition of the Palais–Smale type in the desired function space is not easy due to the presence of the Hardy potential. Related to this fact, the authors of [11,12,13,14] spoke of the existence results of multiple solutions utilizing various critical point theorems in [15,16] without showing the compactness condition of the Palais–Smale type. The existence of at least one nontrivial solution to a nonlinear Dirichlet boundary problem of the elliptic type has been obtained in the paper [11]. Motivated by this paper, Khodabakhshi et al. [13] discovered the existence of at least three different generalized solutions; see [14] for elliptic equations driven by p-Laplacian-like operators. In this circumstance, we also refer to paper [12] for infinitely many solutions.

Recently, in a different approach from [11,12,13,14], Fiscella in [17] dealt with the existence of at least one nontrivial solution to the Schrödinger–Kirchhoff type fractional p-Laplacian with Hardy terms:

$$\left(\alpha +\beta {\left[\varphi \right]}_{s,p}^{p(\theta -1)}\right){(-∆)}_p^s\varphi \left(x\right)+\mathfrak{a}\left(x\right){|\varphi |}^{p-2}v=\lambda {\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}+\gamma g(x,\varphi )\mathrm{in}{ℝ}^N,$$

where $1<p<p^{*}$, $\alpha >0$, $\beta \ge 0$, $\lambda ,\gamma$ are real parameters and a continuous function g verifies the Ambrosetti–Rabinowitz condition. The key tool for obtaining this result is the classical mountain pass theorem. Moreover, by exploiting minimization arguments involving the Nehari manifold and a variational approach, the authors of [18] investigated the existence and multiplicity of positive solutions to the fractional Kirchhoff–Schrödinger equations with singular nonlinearity. Motivated by work [17], the existence results of multiple solutions to Schrödinger–Hardy type equations driven by the nonlocal fractional p-Laplacian

$${\mathcal{L}}_p^s\varphi \left(x\right)+{\mathfrak{a}\left(x\right)|\varphi |}^{p-2}\varphi =\lambda {\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}+\gamma a\left(x\right){|\varphi |}^{r-2}\varphi +\theta g(x,\varphi )\mathrm{in}{ℝ}^N$$

are obtained in [19] by exploiting a variant of Ekeland’s variational principle and the dual fountain theorem as the primary tools. When a Carathéodory function $g:{ℝ}^N\times ℝ\to ℝ$ does not satisfy the Ambrosetti–Rabinowitz condition and a real function $M\in C\left({\mathbb{R}}_0^{+}\right)$ is different from [17,19], the authors of [20] obtained some multiplicity results and uniform boundedness of nontrivial solutions to the p-Laplacian problems of the Kirchhoff–Schrödinger–Hardy type:

$$\begin{array}{c}\hfill -M\left({\int }_{{\mathbb{R}}^N}{|\nabla \varphi |}^{p}dx\right){\mathrm{div}(|\nabla \varphi |}^{p-2}{\nabla \varphi )+\mathfrak{a}\left(x\right)|\varphi |}^{p-2}\varphi =\lambda {\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^p}}+g(x,\varphi )\mathrm{in}{ℝ}^N,\end{array}$$

where a potential function $\mathfrak{a}:{ℝ}^N\to (0,\infty )$ has the condition (V).

Another fascinating aspect of our problem is the presence of Kirchhoff coefficients. The study of the Kirchhoff-type problem, initially suggested by Kirchhoff [21], has a powerful background in diverse applications in physical and biological systems. For this reason, the variational Kirchhoff-type equations have been much studied by many researchers in recent years, for instance, see references [22,23,24,25,26,27,28,29] for problems without the Hardy potential and see also papers [17,20,30,31,32] for problems with the Hardy potential. In this regard, we suppose that the Kirchhoff function $M:[0,\infty )\to {\mathbb{R}}^{+}$ fulfils the conditions as follows:

(M1)

$M\in C\left(\right[0,\infty \left)\right)$ fulfils ${inf}_{\zeta \in {\mathbb{R}}_0^{+}}M\left(\zeta \right)\ge {\alpha }_0>0$, where ${\alpha }_0$ is a constant.

(M2)

There exists a constant $\vartheta \ge 1$ such that

$$\vartheta \mathcal{M}\left(\zeta \right)=\vartheta {\int }_0^{\zeta }M\left(\tau \right)d\tau \ge M\left(\zeta \right)\zeta$$

for $\zeta \ge 0$.

• The authors of [33], by making use of a truncation argument and the mountain pass theorem, derived the existence of nontrivial solutions to a nonlinear elliptic problem with Kirchhoff coefficients when the nondegenerate Kirchhoff function $M:{ℝ}_0^{+}\to {ℝ}_0^{+}$ with (M1) is increasing and continuous, see also [34] and references therein. But, since the increasing condition excludes the non-monotonic case, Pucci–Xiang–Zhang in [27] obtained the multiplicity result for solutions to a class of fractional p-Laplacian problems of the Schrödinger–Kirchhoff type with a more general Kirchhoff function M; namely, the continuous function M fulfils (M1) and the following condition:

($\tilde{M}2$)

There is $\vartheta \in [1,{\displaystyle \frac{p_s^{*}}{p}})$ such that $\vartheta \mathcal{M}\left(\zeta \right)\ge M\left(\zeta \right)\zeta$ for any $\zeta \ge 0$, where $p_s^{*}$ is the fractional critical Sobolev exponent; that is,

$$p_s^{*}:={\displaystyle \frac{Np}{N-sp}}.$$

• A typical model for M satisfying (M1) and ($\tilde{M}2$) is given by $M\left(\zeta \right)=1+\alpha {\zeta }^{\vartheta }$ with $\alpha \ge 0$ for all $\zeta \ge 0$. Hence, this condition includes the non-monotonic case as well as the typical example above. For this reason, in recent years, many researchers have extensively investigated nonlinear elliptic equations in which Kirchhoff coefficients satisfy (M1) and ($\tilde{M}2$); see [22,23,24,27,28,29,35,36]. Inspired by these previous studies, our first results are designed as follows:

Theorem 1. 

Assume that (V), (M1)–(M2), and ($\mathrm{\Psi }1$)–($\mathrm{\Psi }3$) hold. Furthermore, we suppose the following:

(Ψ5)

There are $\mu >p\vartheta$, $T>0$ and a positive constant ${\mathfrak{C}}_0$ such that

$$\mu G(x,\zeta )\le \zeta g(x,\zeta )+{\mathfrak{C}}_0{|\zeta |}^p$$

for all $x\in {\mathbb{R}}^N$ and $|\zeta |\ge T$, where $G(x,\zeta )={\int }_0^{\zeta }g(x,t)dt$.

Then, there is ${\lambda }^{*}>0$ such that problem (1) possesses an unbounded sequence of nontrivial weak solutions $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that $I_{\lambda }\left({\varphi }_n\right)\to \infty$ as $n\to \infty$ for any $\lambda \in (0,{\lambda }^{*})$, where $\mathfrak{E}\left({ℝ}^N\right)$ and $I_{\lambda }$ will be introduced in Section 2.

Theorem 2. 

Suppose that (V), (M1)–(M2), and ($\mathrm{\Psi }1$)–($\mathrm{\Psi }5$) are satisfied. Then, problem (1) admits a sequence of nontrivial solutions $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that $I_{\lambda }\left({\varphi }_n\right)\to 0$ as $n\to \infty$ for any $\lambda \in (0,{\lambda }^{*})$, where ${\lambda }^{*}$ is given in Theorem 1.

To obtain Theorems 1 and 2, the main consideration is to ensure the compactness property of the sequences of the Palais–Smale type. Due to the presence of the Hardy term and Kirchhoff coefficient, problem (1) has more complex nonlinearities than problems without them, so more rigorous analysis must be carried out carefully.

Next, we drive our second main results under a condition on g which is different from ($\mathrm{\Psi }5$). To carry this out, let us assume the following:

(Ψ6)

There is a constant $\nu \ge 1$ such that

$$\nu \mathcal{G}(x,\zeta )\ge \mathcal{G}(x,t\zeta )$$

for $(x,\zeta )\in {ℝ}^N\times ℝ$ and $t\in [0,1]$, where $\mathcal{G}(x,\zeta )=g(x,\zeta )\zeta -p\vartheta G(x,\zeta )$ and $\vartheta$ was given in (M2).

• In fact, the condition ($\mathrm{\Psi }6$) is firstly suggested by the work of Jeanjean [37] in the case of $p=2$. Liu [38] provided the existence of a ground state solution to the p-Laplacian problems with superlinear nonlinearities:

  $$-{\mathrm{div}(|\nabla v|}^{p-2}{\nabla v)+\mathfrak{a}\left(x\right)|v|}^{p-2}v=g(x,v)\mathrm{in}{ℝ}^N.$$

  Here, $\mathfrak{a}\in C\left({\mathbb{R}}^N\right)$ is a potential function satisfying certain conditions and $g:{ℝ}^N\times ℝ\to ℝ$ fulfils the conditions ($\mathrm{\Psi }1$)–($\mathrm{\Psi }2$) and ($\mathrm{\Psi }6$) ($\vartheta =1$). Motivated by this work, Torres in [39] discussed the existence result of a solution to the fractional p-Laplacian problem. Under these conditions, there has been considerable research dealing with the p-Laplacian problem; see [38,40] and see also [36,41,42] for variable exponents $p(·)$. Nevertheless, because the nonlocal Kirchhoff coefficient M exists, the same results cannot be obtained even if we follow a similar argument to [36,38,40,41,42]. More precisely, under the assumptions ($\mathrm{\Psi }1$), ($\mathrm{\Psi }4$), and ($\mathrm{\Psi }6$), we cannot guarantee the compactness condition of the Palais–Smale type when the Kirchhoff coefficient M fulfils the condition (M2). In particular, it is decisive to the fact that $\vartheta \mathcal{M}\left(\zeta \right)-M\left(\zeta \right)\zeta$ is increasing for all $\zeta \ge 0$ to validate this compactness condition of an energy functional corresponding to an elliptic problem with a superlinear term satisfying ($\mathrm{\Psi }6$). Because of this reason, when (M2) is satisfied, many authors considered a condition of the nonlinear term which is different from ($\mathrm{\Psi }6$); see [22,23,24,27,28,29,35,36]. To overcome this difficulty when ($\mathrm{\Psi }6$) is assumed, we suppose that

(M3)

There exists a non-negative constant $\tilde{\mathfrak{K}}$ such that

$$\widehat{\mathcal{M}}\left(\tau \zeta \right)\le \widehat{\mathcal{M}}\left(\zeta \right)+\tilde{\mathfrak{K}}$$

for any $\zeta \ge 0$ and $\tau \in [0,1]$, where $\widehat{\mathcal{M}}\left(\zeta \right)=\vartheta \mathcal{M}\left(\zeta \right)-M\left(\zeta \right)\zeta$.

The condition above originally comes from papers [25,31]. This is different from the assumptions (M2) and ($\tilde{M}2$). For instance, let us consider the function

$$M\left(\zeta \right)=\left(1+{\displaystyle \frac{{\zeta }^2}{\sqrt{1+{\zeta }^4}}}\right)\zeta +{(1+\zeta )}^{-{\displaystyle \frac{5}{2}}}$$

with its primitive function

$$\mathcal{M}\left(\zeta \right)={\displaystyle \frac{1}{2}}\left({\zeta }^2+\sqrt{1+{\zeta }^4}-1\right)-{\displaystyle \frac{2}{3}}{(1+\zeta )}^{-{\displaystyle \frac{3}{2}}}+{\displaystyle \frac{2}{3}}$$

for all $\zeta \ge 0$; then, it is evident that M is not monotone and

$$\widehat{\mathcal{M}}\left(\zeta \right)=\left({\displaystyle \frac{\vartheta }{2}}-1\right){\zeta }^2+{\displaystyle \frac{\vartheta }{2}}\sqrt{1+{\zeta }^4}-\zeta {(1+\zeta )}^{-{\displaystyle \frac{5}{2}}}-{\displaystyle \frac{2\vartheta }{3}}{(1+\zeta )}^{-{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{{\zeta }^4}{\sqrt{1+{\zeta }^4}}}+{\displaystyle \frac{\vartheta }{6}}.$$

If we put $p=2$ and $N=4$ in ($\tilde{M}2$), then we cannot seek a positive real number $\vartheta \in [1,2)$ such that $\widehat{\mathcal{M}}\left(\zeta \right)\ge 0$ for any $\zeta \ge 0$. However, we can look for a non-negative constant $\tilde{\mathfrak{K}}$ ensuring our condition (M3) when $\vartheta \ge 2$.

From this perspective, we demonstrate the existence of a sequence of infinitely many energy solutions to (1) without assuming the monotone condition of $\widehat{\mathcal{M}}\left(\zeta \right)$ when ($\mathrm{\Psi }6$) is supposed. The basic idea of our proof for the existence of multiple small energy solutions comes from the recent studies [24,25,31]. This existence result to nonlinear problems of the elliptic type is specifically motivated by contributions from recent works [26,36,38,40,41,43,44], and the references therein. Our second results are formulated as follows:

Theorem 3. 

Assume that (V), (M1)–(M3), ($\mathrm{\Psi }1$)–($\mathrm{\Psi }3$), and ($\mathrm{\Psi }6$) hold. Then, problem (1) possesses an unbounded sequence of nontrivial weak solutions $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that $I_{\lambda }\left({\varphi }_n\right)\to \infty$ as $n\to \infty$ for any $\lambda \in (-\infty ,0]$.

Theorem 4. 

Assume that (V), (M1)–(M3), ($\mathrm{\Psi }1$)–($\mathrm{\Psi }4$), and ($\mathrm{\Psi }6$) hold. Then, problem (1) admits a sequence of nontrivial solutions $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that $I_{\lambda }\left({\varphi }_n\right)\to 0$ as $n\to \infty$ for any $\lambda \in (-\infty ,0]$.

The outline of this paper is as follows. Section 2 reviews some essential preliminary knowledge of function spaces and then presents the variational framework associated with problem (1) to be utilized in the paper. Also, we provide the fountain theorem and the dual fountain theorem which are the main tools. Section 3 and Section 4 give Theorems 1–2 and Theorems 3–4, respectively, which are existence results of infinitely many nontrivial solutions to problem (1). To carry this out, we first establish the Palais–Smale type compactness condition for the energy function corresponding to our problem under the appropriate conditions for g.

2. Preliminaries and Variational Setting

In this section, we shortly present some useful definitions and fundamental properties of the fractional Sobolev spaces that will be utilized in the present paper. Let $0<s<1<p<+\infty$ be real numbers. Let the fractional Sobolev space $W^{s,p}\left({ℝ}^N\right)$ be defined as follows:

$$W^{s,p}\left({ℝ}^N\right):=\left\{\varphi \in L^p\left({ℝ}^N\right):{\int }_{{ℝ}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi \left(x\right)-\varphi \left(z\right)|}^p}{{|x-z|}^{N+ps}}}dxdz<+\infty \right\},$$

equipped with the norm

$${\Vert \varphi \Vert }_{W^{s,p}\left({ℝ}^N\right)}:={\left({\Vert \varphi \Vert }_{L^p\left({ℝ}^N\right)}^p+{|\varphi |}_{W^{s,p}\left({\mathbb{R}}^N\right)}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

where

$${\Vert \varphi \Vert }_{L^p\left({ℝ}^N\right)}^p:={\int }_{{ℝ}^N}{|\varphi \left(x\right)|}^{p}dx\mathrm{and}{|\varphi |}_{W^{s,p}\left({\mathbb{R}}^N\right)}^p:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi \left(x\right)-\varphi \left(z\right)|}^p}{{|x-z|}^{N+ps}}}dxdz.$$

Then, $W^{s,p}\left({ℝ}^N\right)$ is a reflexive separable Banach space. Also, the space $C_0^{\infty }\left({ℝ}^N\right)$ is dense in $W^{s,p}\left({ℝ}^N\right)$, that is, $W_0^{s,p}\left({ℝ}^N\right)=W^{s,p}\left({ℝ}^N\right)$ (see, e.g., [45,46]).

Lemma 1 

([46,47]). Let $0<s<1<p<+\infty$ with $ps<N$. Then, there is a constant $C>0$ depending on s, p, and N such that

$${\Vert \varphi \Vert }_{L^{p_s^{*}}\left({ℝ}^N\right)}\le C{|\varphi |}_{W^{s,p}\left({ℝ}^N\right)}$$

for all $\varphi \in W^{s,p}\left({ℝ}^N\right)$. Also, the space $W^{s,p}\left({ℝ}^N\right)$ is continuously embedded in $L^{\kappa }\left({ℝ}^N\right)$ for any $\kappa \in [p,p_s^{*}]$. In addition, the embedding

$$W^{s,p}\left({ℝ}^N\right)\hookrightarrow L_{loc}^{\kappa }\left({ℝ}^N\right)$$

is compact for $\kappa \in [p,p_s^{*})$.

Now, let us consider the space $W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)$ defined as follows:

$$W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right):=\left\{\varphi \in L^p\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz<+\infty \right\},$$

where a kernel function $\mathcal{K}:{\mathbb{R}}^N\times {\mathbb{R}}^N\setminus \left\{(0,0)\right\}\to (0,+\infty )$ admits the properties (L1)–(L3). By the condition (L1), the function

$$(x,z)\mapsto (\varphi \left(x\right)-\varphi \left(z\right)){\mathcal{K}}^{{\displaystyle \frac{1}{p}}}(x,z)\in L^p\left({ℝ}^{2N}\right)$$

for any $\varphi \in C_0^{\infty }\left({ℝ}^N\right)$. Let us denote by $W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)$ the completion of $C_0^{\infty }\left({ℝ}^N\right)$ with respect to the norm

$${\Vert \varphi \Vert }_{W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)}:={\left({\Vert \varphi \Vert }_{L^p\left({\mathbb{R}}^N\right)}^p+{|\varphi |}_{W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

where

$${|\varphi |}_{W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)}^p:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz.$$

In what follows, let $0<s<1<p<+\infty$ with $ps<N$ and let the kernel function $K:{ℝ}^N\times {\mathbb{R}}^N\setminus \left\{(0,0)\right\}\to (0,\infty )$ ensure the assumptions (L1)–(L3).

Lemma 2 

([28]). If  $\varphi \in W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)$, then $\varphi \in W^{s,p}\left({ℝ}^N\right)$. Moreover, we have

$${\Vert \varphi \Vert }_{W^{s,p}\left({ℝ}^N\right)}\le max\{1,{\gamma }_0^{-{\displaystyle \frac{1}{p}}}\}{\Vert \varphi \Vert }_{W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)},$$

where ${\gamma }_0$ is given in (L2).

Under the condition (V), let us define the linear subspace of $W_{\mathcal{K}}^{s,p}\left({\mathbb{R}}^N\right)$ as

$$\mathfrak{E}\left({ℝ}^N\right):=\left\{\varphi \in L^p(\mathfrak{a},{ℝ}^N):{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz<+\infty \right\},$$

equipped with the norm

$${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}:={\left({\left[\varphi \right]}_{p,\mathcal{K}}+{\Vert \varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

where

$${\left[\varphi \right]}_{p,\mathcal{K}}:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p{\mathcal{K}(x,z)dxdz\mathrm{and}\Vert \varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p:={\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx.$$

Then, $\mathfrak{E}\left({ℝ}^N\right)$ is continuously embedded into $W^{s,p}\left({\mathbb{R}}^N\right)$ as a closed subspace. Therefore, $(\mathfrak{E}\left({ℝ}^N\right),\Vert ·{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)})$ is also a separable reflexive Banach space.

From Lemmas 1 and 2, we can obtain the following assertion immediately.

Lemma 3 

([28]). For any $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$ and $1\le q\le p_s^{*}$, there exists a positive constant $C_0=C_0(N,p,s)$ such that

$$\begin{array}{cc}\hfill {\Vert \varphi \Vert }_{L^q\left({ℝ}^N\right)}^p& \le C_0{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi \left(x\right)-\varphi \left(z\right)|}^p}{{|x-z|}^{N+ps}}}dxdz\hfill \\ \hfill & \le {\displaystyle \frac{C_0}{{\gamma }_0}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz,\hfill \end{array}$$

where ${\gamma }_0$ is given in (L2). Consequently, the space $\mathfrak{E}\left({ℝ}^N\right)$ is continuously embedded in $L^q\left({ℝ}^N\right)$ for any $q\in [p,p_s^{*}]$. In addition, the embedding

$$\mathfrak{E}\left({ℝ}^N\right)\hookrightarrow L^q\left({ℝ}^N\right)$$

is compact for $q\in [p,p_s^{*})$.

The following consequence is the fractional Hardy inequality which is a useful tool given in [48].

Lemma 4. 

For any $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$, in case $sp<N$, and for all $\varphi \in \mathfrak{E}\left({ℝ}^N\right)\setminus \left\{0\right\}$, in case $sp>N$, there is a positive constant ${\mathcal{C}}_{\mathcal{H}}:=\mathcal{C}(N,s,p)$ such that

$$\begin{array}{cc}\hfill {\Vert \varphi \Vert }_{H_p}^p:={\int }_{{ℝ}^N}{\displaystyle \frac{{|\varphi \left(x\right)|}^p}{{|x|}^{sp}}}dx& \le {\mathcal{C}}_{\mathcal{H}}{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{\displaystyle \frac{{|\varphi \left(x\right)-\varphi \left(z\right)|}^p}{{|x-z|}^{N+sp}}}dxdz,\hfill \\ \hfill & \le {\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}}{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz.\hfill \end{array}$$

Definition 1. 

A function $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$ is a weak solution of problem (1) if the equality

$$\begin{array}{cc}\hfill M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)& {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))(v\left(x\right)-v\left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p-2}\varphi vdx=\lambda {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}vdx+{\int }_{{ℝ}^N}g(x,\varphi )vdx\hfill \end{array}$$

holds for every $v\in \mathfrak{E}\left({ℝ}^N\right)$.

Let us define the functional $\mathrm{\Phi }:X\to ℝ$ by

$$\mathrm{\Phi }\left(\varphi \right)={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx.$$

Then, it is effortless to prove that $\mathrm{\Phi }$ is well defined on $\mathfrak{E}\left({ℝ}^N\right)$, $\mathrm{\Phi }\in C^1(\mathfrak{E}\left({ℝ}^N\right),ℝ)$, and its Fréchet derivative is given by

$$\begin{array}{cc}\hfill \langle {\mathrm{\Phi }}^{\prime }\left(\varphi \right),v\rangle =M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)& {\int }_{{ℝ}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))(v\left(x\right)-v\left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p-2}\varphi vdx\hfill \end{array}$$

for any $\varphi ,v\in \mathfrak{E}\left({ℝ}^N\right)$, where $〈·,·〉$ denotes the pairing of $\mathfrak{E}\left({ℝ}^N\right)$ and its dual $\mathfrak{E}{\left({ℝ}^N\right)}^{*}$. Define the functional ${\mathrm{\Psi }}_{\lambda }:\mathfrak{E}\left({ℝ}^N\right)\to ℝ$ by

$${\mathrm{\Psi }}_{\lambda }\left(\varphi \right)={\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx+{\int }_{{ℝ}^N}G(x,\varphi )dx.$$

Then, ${\mathrm{\Psi }}_{\lambda }\in C^1(\mathfrak{E}\left({ℝ}^N\right),ℝ)$ and its Fréchet derivative is

$$〈{\mathrm{\Psi }}_{\lambda }^{\prime }\left(\varphi \right),v〉:=\lambda {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}vdx+{\int }_{{ℝ}^N}g(x,\varphi )vdx$$

for any $\varphi ,v\in \mathfrak{E}\left({ℝ}^N\right)$. Next, the functional $I_{\lambda }:\mathfrak{E}\left({ℝ}^N\right)\to ℝ$ is defined by

$$I_{\lambda }\left(\varphi \right)=\mathrm{\Phi }\left(\varphi \right)-{\mathrm{\Psi }}_{\lambda }\left(\varphi \right).$$

Then, $I_{\lambda }\in C^1(\mathfrak{E}\left({ℝ}^N\right),{ℝ}^N)$ and its Fréchet derivative is

$$\begin{array}{cc}\hfill 〈I_{\lambda }^{\prime }\left(\varphi \right),v〉& =M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right){\int }_{{ℝ}^N}{\int }_{{\mathbb{R}}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))(v\left(x\right)-v\left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p-2}\varphi vdx-\lambda {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}vdx-{\int }_{{ℝ}^N}g(x,\varphi )vdx\hfill \end{array}$$

(2)

for any $\varphi ,v\in \mathfrak{E}\left({ℝ}^N\right)$.

Let $\mathfrak{E}$ be a separable and reflexive Banach space. Then, it is known (see [49,50]) that there are $\left\{e_n\right\}\subseteq \mathfrak{E}$ and $\left\{{\hslash }_n^{*}\right\}\subseteq {\mathfrak{E}}^{*}$ such that

$$\mathfrak{E}=\overline{\mathrm{span}\{e_n:n=1,2,\cdots \}},{\mathfrak{E}}^{*}=\overline{\mathrm{span}\{h_n^{*}:n=1,2,\cdots \}},$$

and

$$\begin{array}{c}\hfill 〈{\hslash }_i^{*},e_j〉=\left\{\begin{array}{cc}1\hfill & ifi=j\hfill \\ 0\hfill & ifi\ne j.\hfill \end{array}\right.\end{array}$$

Let us denote ${\mathfrak{E}}_n=\mathrm{span}\left\{e_n\right\}$, ${\mathfrak{Y}}_k={⨁}_{n=1}^k{\mathfrak{E}}_n$, and ${\mathfrak{Z}}_k=\overline{{⨁}_{n=k}^{\infty }{\mathfrak{E}}_n}$.

Definition 2. 

Suppose that $(\mathfrak{E},\Vert ·\Vert )$ is a real Banach space. Then, the functional $\mathcal{F}$ ensures the Cerami condition at level $c\in ℝ$ (${\left(C\right)}_c$-condition for short) in $\mathfrak{E}$, if any ${\left(C\right)}_c$-sequence ${\left\{{\varphi }_n\right\}}_{n\in \mathbb{N}}\subset \mathfrak{E}$, i.e., $\mathcal{F}\left({\varphi }_n\right)\to c$ and $\Vert {\mathcal{F}}^{\prime }\left({\varphi }_n\right){\Vert }_{{\mathfrak{E}}^{*}}(1+\Vert {\varphi }_n{\Vert }_{\mathfrak{E}})\to 0\mathrm{as}n\to \infty$, admits a convergent subsequence in $\mathfrak{E}$.

Lemma 5 

([41,51]). Suppose that $(\mathfrak{E},\Vert ·\Vert )$ is a separable reflexive Banach space, and that the functional $\mathcal{F}\in C^1(\mathfrak{E},ℝ)$ is even and ensures the ${\left(C\right)}_c$-condition for any $c>0$. If, for each large enough $k\in N$, there are ${\beta }_k>{\alpha }_k>0$ such that

(1) 

$b_k:={inf\{\mathcal{F}\left(\phi \right):\Vert \phi \Vert }_{\mathfrak{E}}={\alpha }_k,\phi \in {\mathfrak{Z}}_k\}\to \infty \mathrm{as}k\to \infty$;

(2) 

$a_k:={max\{\mathcal{F}\left(\phi \right):\Vert \phi \Vert }_{\mathfrak{E}}={\beta }_k,\phi \in {\mathfrak{Y}}_k\}\le 0,$

then there is a sequence $\left\{{\varphi }_n\right\}\subset \mathfrak{E}$ such that ${\mathcal{F}}^{\prime }\left({\varphi }_n\right)=0$ and $\mathcal{F}\left({\varphi }_n\right)\to +\infty$ as $n\to +\infty$.

Definition 3. 

Suppose that $(\mathfrak{E},\Vert ·\Vert )$ is a real separable reflexive Banach space, $\mathcal{F}\in C^1(\mathfrak{E},ℝ)$, $c\in ℝ$. Then, $\mathcal{F}$ fulfils the ${\left(C\right)}_c^{*}$-condition (with respect to ${\mathfrak{Y}}_n$) if any sequence ${\left\{{\varphi }_n\right\}}_{n\in N}\subset \mathfrak{E}$ for which ${\varphi }_n\in {\mathfrak{Y}}_n$, for any $n\in N$,

$$\mathcal{F}\left({\varphi }_n\right)\to cand\Vert {\left(\mathcal{F}{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right){\Vert }_{{\mathfrak{E}}^{*}}(1+\Vert {\varphi }_n\Vert )\to 0\mathrm{as}n\to \infty ,$$

possesses a subsequence converging to a critical point of $\mathcal{F}$.

Lemma 6 

([44]). Suppose that $(\mathfrak{E},\Vert ·\Vert )$ is a real reflexive and separable Banach space, and $\mathcal{F}\in C^1(\mathfrak{E},ℝ)$ is an even functional. If there is $k_0>0$ such that, for each $k\ge k_0$, there exist ${\beta }_k>{\alpha }_k>0$ such that

(D1) 

$inf\{\mathcal{F}\left(\phi \right):\Vert \phi {\Vert }_{\mathfrak{E}}={\beta }_k,\phi \in {\mathfrak{Z}}_k\}\ge 0$;

(D2) 

$b_k:={max\{\mathcal{F}\left(\phi \right):\Vert \phi \Vert }_{\mathfrak{E}}={\alpha }_k,\phi \in {\mathfrak{Y}}_k\}<0$;

(D3) 

$c_k:={inf\{\mathcal{F}\left(\phi \right):\Vert \phi \Vert }_{\mathfrak{E}}\le {\beta }_k,\phi \in {\mathfrak{Z}}_k\}\to 0$ as $k\to \infty$;

(D4) 

$\mathcal{F}$ satisfies the ${\left(C\right)}_c^{*}$-condition for every $c\in [c_{k_0},0)$,

then $\mathcal{F}$ admits a sequence of negative critical values $c_k<0$ converging to 0.

3. Proof of Theorems 1 and 2

In this section, we present the proof of Theorems 1 and 2. To carry this out, we show that the energy functional $I_{\lambda }$ verifies the Cerami condition. This plays a crucial role in demonstrating the existence of nontrivial weak solutions for the given problem. The fundamental idea of proof of this consequence follows similar arguments to those in [17]; see also [19,20]. However, more complicated analyses than those of the previous works [17,19] must be performed meticulously because of the presence of a nonlocal Kirchhoff coefficient M.

Lemma 7. 

Assume that (V),(M1)–(M2), and ($\mathrm{\Psi }1$)–($\mathrm{\Psi }2$),($\mathrm{\Psi }5$) hold. Then, there is ${\lambda }^{*}>0$ such that the functional $I_{\lambda }$ fulfils the ${\left(C\right)}_c$-condition for any $\lambda \in (-\infty ,{\lambda }^{*})$.

Proof. 

Let $\left\{{\varphi }_n\right\}$ be a ${\left(C\right)}_c$-sequence in $\mathfrak{E}\left({ℝ}^N\right)$, namely,

$$I_{\lambda }\left({\varphi }_n\right)\to c\mathrm{and}\Vert I_{\lambda }^{\prime }\left({\varphi }_n\right){\Vert }_{\mathfrak{E}{\left({ℝ}^N\right)}^{*}}(1+\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)})\to 0,\mathrm{as}n\to \infty ,$$

(3)

where c is any real number.

Now, we show that $\left\{{\varphi }_n\right\}$ is bounded in $\mathfrak{E}\left({ℝ}^N\right)$. To this end, suppose to the contrary that $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}>1$ and $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty$ as $n\to \infty$. Let ${\phi }_n={\varphi }_n/\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}$. Then, $\Vert {\phi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}=1$. Passing to a subsequence, we may assume that ${\phi }_n\rightharpoonup \phi$ as $n\to \infty$ in $\mathfrak{E}\left({ℝ}^N\right)$; then, by virtue of Lemma 3,

$${\phi }_n\to \phi \mathrm{in}{L}^m\left({ℝ}^N\right)\mathrm{and}{\phi }_n\left(x\right)\to \phi \left(x\right)\mathrm{a}.\mathrm{e}.\mathrm{in}{ℝ}^N$$

(4)

with $m\in [p,p^{*})$. Since $\mathfrak{a}\left(x\right)\to +\infty$ as $|x|\to \infty$, it follows from the analogous argument to that in [24] that

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{\vartheta p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx& -C_1{\int }_{\{|{\varphi }_n|\le T\}}\left(|{\varphi }_n{|}^p+{\sigma }_0\left(x\right)|{\varphi }_n|+{\sigma }_1{|{\varphi }_n|}^q\right)dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{\vartheta p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\mathfrak{C}}_1\hfill \end{array}$$

(5)

for some positive constant ${\mathfrak{C}}_1$ and for any positive constant $C_1$.

Set

$${\lambda }^{*}={\displaystyle \frac{min\{{\alpha }_0,1\}{\gamma }_0(\mu -\vartheta p)}{\vartheta {\mathcal{C}}_{\mathcal{H}}(\mu -p)}},$$

(6)

where ${\mathcal{C}}_{\mathcal{H}}$, ${\gamma }_0$, ${\alpha }_0$, and $\vartheta$ are given in Lemma 4, (L2), (M1), and (M2), respectively. Then, it follows from (3), (5), (M1)–(M2), ($\mathrm{\Psi }1$), and ($\mathrm{\Psi }5$) that, for any $\lambda \in (0,{\lambda }^{*})$ and for n large enough,

$$\begin{array}{cc}\hfill c+1& \ge I_{\lambda }\left({\varphi }_n\right)-{\displaystyle \frac{1}{\mu }}〈I_{\lambda }^{\prime }\left({\varphi }_n\right),{\varphi }_n〉\hfill \\ \hfill & ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & -{\displaystyle \frac{1}{\mu }}M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\int }_{{\mathbb{R}}^N}{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & -{\displaystyle \frac{1}{\mu }}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+{\displaystyle \frac{\lambda }{\mu }}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx+{\displaystyle \frac{1}{\mu }}{\int }_{{ℝ}^N}g(x,{\varphi }_n){\varphi }_{n}dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p\vartheta }}M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\int }_{{ℝ}^N}{\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & -{\displaystyle \frac{1}{\mu }}M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\int }_{{ℝ}^N}{\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & -{\displaystyle \frac{1}{\mu }}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+{\displaystyle \frac{\lambda }{\mu }}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx+{\displaystyle \frac{1}{\mu }}{\int }_{{ℝ}^N}g(x,{\varphi }_n){\varphi }_{n}dx\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\int }_{{\mathbb{R}}^N}{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+{\displaystyle \frac{1}{\mu }}{\int }_{{\mathbb{R}}^N}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx\hfill \\ \hfill & -\lambda \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & \ge {\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+{\displaystyle \frac{1}{\mu }}{\int }_{\{|{\varphi }_n|\le T\}}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx\hfill \\ \hfill & +{\displaystyle \frac{1}{\mu }}{\int }_{\{|{\varphi }_n|\ge T\}}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx-\lambda \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & \ge {\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-C_1{\int }_{\{|{\varphi }_n|\le T\}}\left(|{\varphi }_n{|}^p+{\sigma }_0\left(x\right)|{\varphi }_n|+{\sigma }_1{|{\varphi }_n|}^q\right)dx\hfill \\ \hfill & -{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}{\int }_{\{|{\varphi }_n|\ge T\}}{|{\varphi }_n|}^{p}dx\hfill \\ \hfill & \ge \left[{\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)-{\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right)\right]{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)|{\varphi }_n{|}^{p}dx-{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}{\int }_{{ℝ}^N}{|{\varphi }_n|}^{p}dx-{\mathfrak{C}}_1\hfill \\ \hfill & \ge min\left\{{\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)-{\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right),{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right)\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p\hfill \\ \hfill & -{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}\Vert {\varphi }_n{\Vert }_{L^p\left({ℝ}^N\right)}^p-{\mathfrak{C}}_1.\hfill \end{array}$$

Thus, we arrive at

$$\begin{array}{cc}\hfill c+1+{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}& \Vert {\varphi }_n{\Vert }_{L^p\left({ℝ}^N\right)}^p+{\mathfrak{C}}_1\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p\mu }}min\left\{{\alpha }_0\left(\mu -p\vartheta \right)-{\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}}\left(\mu -p\right),{\displaystyle \frac{1}{2}}\left(\mu -p\right)\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p.\hfill \end{array}$$

Let us divide this by ${\displaystyle \frac{1}{p\mu }}min\left\{{\alpha }_0\left(\mu -p\vartheta \right)-{\displaystyle \frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}}\left(\mu -p\right),{\displaystyle \frac{1}{2}}\left(\mu -p\right)\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p$. With the aid of relation (4), taking the limit supremum of this inequality as $n\to \infty$ yields that

$$\begin{array}{cc}\hfill 1& \le {\displaystyle \frac{p{\mathfrak{C}}_0}{min\left\{{\alpha }_0\left(\mu -p\vartheta \right)-\frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}\left(\mu -p\right),\frac{1}{2}\left(\mu -p\right)\right\}}}\underset{n\to \infty }{lim\; sup}\Vert {\phi }_n{\Vert }_{L^p\left({ℝ}^N\right)}^p\hfill \\ \hfill & ={\displaystyle \frac{p{\mathfrak{C}}_0}{min\{min\left\{{\alpha }_0\left(\mu -p\vartheta \right)-\frac{{\mathcal{C}}_{\mathcal{H}}\lambda }{{\gamma }_0}\left(\mu -p\right),\frac{1}{2}\left(\mu -p\right)\right\}}}{\Vert \phi \Vert }_{L^p\left({ℝ}^N\right)}^p.\hfill \end{array}$$

(7)

Next, if $\lambda \in (-\infty ,0]$, then it follows from (M1)–(M2), ($\mathrm{\Psi }1$), ($\mathrm{\Psi }5$), and (5) that

$$\begin{array}{cc}\hfill c+1& \ge I_{\lambda }\left({\varphi }_n\right)-{\displaystyle \frac{1}{\mu }}〈I_{\lambda }^{\prime }\left({\varphi }_n\right),{\varphi }_n〉\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-\lambda \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & +{\displaystyle \frac{1}{\mu }}{\int }_{{\mathbb{R}}^N}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx\hfill \\ \hfill & \ge {\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & +{\displaystyle \frac{1}{\mu }}{\int }_{\{|{\varphi }_n|\le T\}}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx\hfill \\ \hfill & +{\displaystyle \frac{1}{\mu }}{\int }_{\{|{\varphi }_n|\ge T\}}\left(g(x,{\varphi }_n){\varphi }_n-\mu G(x,{\varphi }_n)\right)dx\hfill \\ \hfill & \ge {\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -C_1{\int }_{\{|{\varphi }_n|\le T\}}\left({\sigma }_0\left(x\right)|{\varphi }_n|+{\sigma }_1{|{\varphi }_n|}^q\right)dx-{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}{\int }_{\{|{\varphi }_n|\ge T\}}{|{\varphi }_n|}^{p}dx\hfill \\ \hfill & \ge {\alpha }_0\left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)|{\varphi }_n{|}^{p}dx-{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}{\int }_{{ℝ}^N}{|{\varphi }_n|}^{p}dx-{\mathfrak{C}}_1\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)min\{{\alpha }_0,{\displaystyle \frac{1}{2}}\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}\Vert {\varphi }_n{\Vert }_{L^p\left({ℝ}^N\right)}^p-{\mathfrak{C}}_1\hfill \end{array}$$

for sufficiently large n. Hence, we deduce that

$$\begin{array}{c}\hfill c+1+{\displaystyle \frac{{\mathfrak{C}}_0}{\mu }}\Vert {\varphi }_n{\Vert }_{L^p\left({ℝ}^N\right)}^p+{\tilde{\mathfrak{K}}}_0\ge \left({\displaystyle \frac{1}{p\vartheta }}-{\displaystyle \frac{1}{\mu }}\right)min\{{\alpha }_0,{\displaystyle \frac{1}{2}}\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p.\end{array}$$

Let us divide this by $\left({\displaystyle \frac{\mu -p\vartheta }{p\mu \vartheta }}-{\displaystyle \frac{1}{\mu }}\right)min\{{\alpha }_0,{\displaystyle \frac{1}{2}}\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p$. With the help of relation (4), taking the limit supremum of this inequality as $n\to \infty$ yields that

$$1\le {\displaystyle \frac{{\mathfrak{C}}_{0}p\vartheta }{(\mu -p\vartheta )min\{{\alpha }_0,\frac{1}{2}\}}}{\Vert \phi \Vert }_{L^p\left({ℝ}^N\right)}^p.$$

(8)

Hence, taking into account (7) and (8), we achieve that $\phi \ne 0$ when $\lambda \in (-\infty ,{\lambda }^{*})$.

Next, we will result in a contradiction by showing the fact that $\phi \left(x\right)=0$ for almost all $x\in {\mathbb{R}}^N$. To carry this out, we first take into account two claims as follows:

Claim 1. For any $\lambda \in (-\infty ,{\lambda }^{*})$, one has

$$\begin{array}{c}\hfill {\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\to \infty \mathrm{as}n\to \infty .\end{array}$$

(9)

Indeed, let us consider $\lambda \in (0,{\lambda }^{*})$. Since $\lambda <{\lambda }^{*}<{\displaystyle \frac{{\gamma }_0{\alpha }_0}{{\mathcal{C}}_{\mathcal{H}}\vartheta }}$, taking into account Lemma 4, (M1)–(M2), we have

$$\begin{array}{cc}\hfill I_{\lambda }\left({\varphi }_n\right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{{\alpha }_0}{p\vartheta }}{\int }_{{\mathbb{R}}^N}|{\varphi }_n\left(x\right)-{\varphi }_n{\left(z\right)|}^p\mathcal{K}(x,z)dxdz+{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{p{\gamma }_0}}{\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz-{\int }_{{\mathbb{R}}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}\left({\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}}\right){\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{\mathbb{R}}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx.\hfill \end{array}$$

(10)

By (10), one has

$$\begin{array}{cc}\hfill {\int }_{{ℝ}^N}G(x,{\varphi }_n)dx& \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-I_{\lambda }\left({\varphi }_n\right).\hfill \end{array}$$

(11)

Also, if $\lambda \in (-\infty ,0]$, by taking into account (M1)–(M2), we have

$$\begin{array}{cc}\hfill I_{\lambda }\left({\varphi }_n\right)& \ge {\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{{\alpha }_0}{p\vartheta }}{\int }_{{\mathbb{R}}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{\mathbb{R}}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx.\hfill \end{array}$$

(12)

Using (12), we obtain

$$\begin{array}{cc}\hfill {\int }_{{ℝ}^N}G(x,{\varphi }_n)dx& \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-I_{\lambda }\left({\varphi }_n\right).\hfill \end{array}$$

(13)

Since $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty$ as $n\to \infty$, we assert by (11) and (13) that (13) is claimed.

Claim 2. For any $\lambda \in (-\infty ,{\mu }^{*})$, we have

$$\begin{array}{cc}\hfill 3^{\vartheta }& max\left\{\mathcal{M}\left(1\right),{\displaystyle \frac{min\{m_0,1\}(\nu -\vartheta p)}{\vartheta (\nu -p)}}-{\displaystyle \frac{\lambda }{H_p}},1\right\}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}\left({\int }_{{ℝ}^N}G(x,{\varphi }_n)dx+I_{\lambda }\left({\varphi }_n\right)\right).\hfill \end{array}$$

(14)

In fact, note that $\mathcal{M}\left(\zeta \right)\le \mathcal{M}\left(1\right)\left(1+{\zeta }^{\vartheta }\right)$ for all $\zeta \in ℝ$ because, if $0\le \zeta <1$, then $\mathcal{M}\left(\zeta \right)={\int }_0^{\zeta }M\left(\tau \right)d\tau \le \mathcal{M}\left(1\right)$ and, if $\zeta >1$, then $\mathcal{M}\left(\zeta \right)\le \mathcal{M}\left(1\right){\zeta }^{\vartheta }$. First, let us consider the case $\mu \in (0,{\mu }^{*})$. Then, this together with the assumption (M2) yields that

$$\begin{array}{cc}\hfill I_{\lambda }\left({\varphi }_n\right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & \le \mathcal{M}\left(1\right)\left[1+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}^{\vartheta }\right]+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le max\left\{\mathcal{M}\left(1\right),{\displaystyle \frac{1}{p}}\right\}\left(1+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}^{\vartheta }+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\right)-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le max\left\{\mathcal{M}\left(1\right),{\displaystyle \frac{1}{p}}\right\}{\left(1+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\right)}^{\vartheta }-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le 2^{\vartheta }max\left\{\mathcal{M}\left(1\right),{\displaystyle \frac{1}{p}}\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx.\hfill \end{array}$$

(15)

Also, if $\lambda \in (-\infty ,0]$, the assumption (M2) implies that

$$\begin{array}{cc}\hfill I_{\lambda }\left({\varphi }_n\right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & \le \mathcal{M}\left(1\right)\left[1+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}^{\vartheta }\right]-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}}{\left[{\varphi }_n\right]}_{p,\mathcal{K}}+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},{\displaystyle \frac{1}{p}}\right\}\left[1+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}^{\vartheta }+{\left[{\varphi }_n\right]}_{p,\mathcal{K}}+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\right]\hfill \\ \hfill & -{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\left(1+2{\left[{\varphi }_n\right]}_{p,\mathcal{K}}+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\right)}^{\vartheta }\hfill \\ \hfill & -{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & \le 3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx.\hfill \end{array}$$

(16)

Then, we obtain by the relations (15) and (16) that

$$3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}\ge {\displaystyle \frac{1}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}\left({\int }_{{ℝ}^N}G(x,{\varphi }_n)dx+I_{\lambda }\left({\varphi }_n\right)\right)$$

(17)

for any $\lambda \in (-\infty ,{\lambda }^{*})$.

By using Claims 1 and 2, we will derive a contradiction by showing that $\phi \left(x\right)=0$ for almost all $x\in {\mathbb{R}}^N$. To carry this out, we set ${\mathrm{\Gamma }}_1=\left\{x\in {ℝ}^N:\phi \left(x\right)\ne 0\right\}$. Assume that $\mathrm{meas}\left({\mathrm{\Gamma }}_1\right)\ne 0$. By the convergence (4), we obtain that $|{\varphi }_n\left(x\right)| = |{\phi }_n\left(x\right)|\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty$ as $n\to \infty$ for almost all $x\in {\mathrm{\Gamma }}_1$. In addition, due to ($\mathrm{\Psi }2$), one has

$$\underset{n\to \infty }{lim}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}=\underset{n\to \infty }{lim}{\displaystyle \frac{G(x,{\varphi }_n)}{|{\varphi }_n{|}^{p\vartheta }}}{|{\phi }_n|}^{p\vartheta }=\infty$$

(18)

for all $x\in {\mathrm{\Gamma }}_1$. According to (11)–(18) and Fatou’s lemma, we deduce that

$$\begin{array}{cc}\hfill 3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}& =\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-\frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0},1\right\}{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx}{{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx+I_{\lambda }\left({\varphi }_n\right)}}\hfill \\ & \ge \underset{n\to \infty }{lim\; inf}{\int }_{{ℝ}^N}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx\hfill \\ \hfill & \ge {\int }_{{\mathrm{\Gamma }}_1}\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx\hfill \\ \hfill & ={\int }_{{\mathrm{\Gamma }}_1}\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{G(x,{\varphi }_n)}{|{\varphi }_n{|}^{p\vartheta }}}{|{\phi }_n|}^{p\vartheta }dx=\infty ,\hfill \end{array}$$

which is a contradiction. Thus, $\phi \left(x\right)=0$ for almost all $x\in {ℝ}^N$. As a result, this leads to a contradiction and thus $\left\{{\varphi }_n\right\}$ is bounded in $\mathfrak{E}\left({ℝ}^N\right)$.

By the reflexivity of $\mathfrak{E}\left({ℝ}^N\right)$ and Lemmas 3 and 4, there exists a subsequence, still denoted by $\left\{{\varphi }_n\right\}$, and $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$ such that

$$\begin{array}{cc}\hfill {\varphi }_n\rightharpoonup \varphi \mathrm{in}\mathfrak{E}\left({ℝ}^N\right),& {\varphi }_n\rightharpoonup \varphi \mathrm{in}{L}^p({ℝ}^N{,|x|}^{-sp}),\hfill \\ \hfill \Vert {\varphi }_n-\varphi {\Vert }_{H_p}^p\to \ell ,& {\varphi }_n\to \varphi \mathrm{in}{L}^m\left({ℝ}^N\right),{\varphi }_n\left(x\right)\to \varphi \left(x\right)\mathrm{for}\mathrm{almost}\mathrm{all}x\in {ℝ}^N,\hfill \end{array}$$

(19)

as $n\to \infty$, with $m\in [p,p^{*})$. Then, the sequence

$${\left\{|{\varphi }_n\left(x\right)-{\varphi }_n{\left(z\right)|}^{p-2}({\varphi }_n\left(x\right)-{\varphi }_n\left(z\right))\mathcal{K}{(x,z)}^{{\displaystyle \frac{1}{p^{\prime }}}}\right\}}_{n\in N}$$

is bounded in $L^{p^{\prime }}({\mathbb{R}}^N\times {\mathbb{R}}^N)$, as well as a.e. in ${\mathbb{R}}^N\times {\mathbb{R}}^N$

$$\begin{array}{cc}\hfill & {\mathcal{B}}_n(x,z):={|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^{p-2}({\varphi }_n\left(x\right)-{\varphi }_n\left(z\right))\mathcal{K}{(x,z)}^{{\displaystyle \frac{1}{p^{\prime }}}}\hfill \\ \hfill & \to \mathcal{B}(x,z):={|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))\mathcal{K}{(x,z)}^{{\displaystyle \frac{1}{p^{\prime }}}}\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus, going if necessary to a further subsequence, we infer that ${\mathcal{B}}_n\rightharpoonup \mathcal{B}$ in $L^{p^{\prime }}({\mathbb{R}}^N\times {\mathbb{R}}^N)$ as $n\to \infty$. Furthermore, $|{\varphi }_n{|}^{p-2}{\varphi }_n\rightharpoonup {|\varphi |}^{p-2}\varphi$ in $L^{p^{\prime }}(\mathfrak{a},{ℝ}^N)$. Hence, we assert that, for any $\psi \in \mathfrak{E}\left({ℝ}^N\right)$,

$$\begin{array}{cc}\hfill & {\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^{p-2}({\varphi }_n\left(x\right)-{\varphi }_n\left(z\right))(\psi \left(x\right)-\psi \left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & \to {\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^{p-2}(\varphi \left(x\right)-\varphi \left(z\right))(\psi \left(x\right)-\psi \left(z\right))\mathcal{K}(x,z)dxdz\hfill \end{array}$$

(20)

and

$${\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)|{\varphi }_n\left(x\right){|}^{p-2}{\varphi }_n\left(x\right)\psi \left(x\right)dx\to {\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi \left(x\right)|}^{p-2}\varphi \left(x\right)\psi \left(x\right)dx$$

(21)

as $n\to \infty$, because $(x,z)\mapsto {|\psi \left(x\right)-\psi \left(z\right)|·|x-z|}^{-(n+ps)/p}\in L^p({\mathbb{R}}^N\times {\mathbb{R}}^N)$ and $\psi \in L^p(\mathfrak{a},{ℝ}^N)$.

On the other hand, the sequence

$${\left\{{\displaystyle \frac{|{\varphi }_n{\left(x\right)|}^{p-2}{\varphi }_n\left(x\right)}{{|x|}^{\frac{sp}{p^{\prime }}}}}\right\}}_{n\in N}$$

is bounded in $L^{p^{\prime }}\left({ℝ}^N\right)$, as well as a.e. in ${ℝ}^N$:

$${\displaystyle \frac{|{\varphi }_n{\left(x\right)|}^{p-2}{\varphi }_n\left(x\right)}{{|x|}^{\frac{sp}{p^{\prime }}}}}\to {\displaystyle \frac{{|\varphi \left(x\right)|}^{p-2}\varphi \left(x\right)}{{|x|}^{\frac{sp}{p^{\prime }}}}}\mathrm{as}n\to \infty$$

By (19), we have

$$|{\varphi }_n{|}^{p-2}{\varphi }_n\rightharpoonup {|\varphi |}^{p-2}v\mathrm{in}{L}^{p^{\prime }}({ℝ}^N{,|x|}^{-sp}),$$

so that

$$\underset{n\to \infty }{lim}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^{p-2}{\varphi }_n}{{|x|}^{sp}}}\psi dx={\int }_{{ℝ}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}\psi dx$$

(22)

for any $\psi \in \mathfrak{E}\left({ℝ}^N\right)$.

From (20)–(22), we derive that

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^{p-2}({\varphi }_n\left(x\right)-{\varphi }_n\left(z\right))(\varphi \left(x\right)-\varphi \left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & ={\left[\varphi \right]}_{p,\mathcal{K}}^p\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)|{\varphi }_n{|}^{p-2}{\varphi }_n\varphi dx={\Vert \varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p,\hfill \\ \hfill & \underset{n\to \infty }{lim}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^{p-2}{\varphi }_n}{{|x|}^{sp}}}\varphi dx={\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx.\hfill \end{array}$$

(24)

Let $\psi \in \mathfrak{E}\left({ℝ}^N\right)$ be fixed and let ${\tilde{\mathrm{\Psi }}}_{\psi }$ denote the linear functional on $\mathfrak{E}\left({ℝ}^N\right)$ with

$${\tilde{\mathrm{\Psi }}}_{\psi }\left(\varphi \right)={\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\psi \left(x\right)-\psi \left(z\right)|}^{p-2}(\psi \left(x\right)-\psi \left(z\right))(\varphi \left(x\right)-\varphi \left(z\right))\mathcal{K}(y,z)dydz$$

for all $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$. Obviously, by the Hölder inequality, ${\tilde{\mathrm{\Psi }}}_{\psi }$ is also continuous, as

$$\begin{array}{cc}\hfill |{\tilde{\mathrm{\Psi }}}_{\psi }\left(\varphi \right)|& \le {\left({\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\psi \left(x\right)-\psi \left(z\right)|}^p\mathcal{K}(x,z)dxdz\right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ \hfill & \times {\left({\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|\psi \left(x\right)-\psi \left(z\right)|}^p\mathcal{K}(x,z)dxdz\right)}^{{\displaystyle \frac{p-1}{p}}}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\hfill \end{array}$$

(25)

for any $\varphi \in \mathfrak{E}\left({ℝ}^N\right)$. Hence, relation (25) yields

$$\underset{n\to \infty }{lim}\left[M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)-M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)\right]{\tilde{\mathrm{\Psi }}}_{\varphi }({\varphi }_n-\varphi )=0,$$

(26)

because the sequence ${\left\{M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)-M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)\right\}}_{n\in \mathbb{N}}$ is bounded in ${ℝ}^N$.

By considering also Lemma 3, (19), the assumption ($\mathrm{\Psi }1$), and the Hölder inequality, we obtain

$$\begin{array}{cc}\hfill & |{\int }_{{ℝ}^N}g(x,{\varphi }_n)({\varphi }_n-\varphi )dx|\hfill \\ \hfill & \le {\int }_{{ℝ}^N}\left({\sigma }_0+{\sigma }_1{|{\varphi }_n|}^{q-1}\right)|{\varphi }_n-\varphi |dx\hfill \\ \hfill & \le \Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}\Vert {\varphi }_n-{\varphi \Vert }_{L^{ℝ}\left({\mathbb{R}}^N\right)}+{\sigma }_1\Vert {\varphi }_n{\Vert }_{L^q\left({\mathbb{R}}^N\right)}^{q-1}\Vert {\varphi }_n-\varphi {\Vert }_{L^q\left({\mathbb{R}}^N\right)}\to 0\hfill \end{array}$$

(27)

as $n\to \infty$.

Let us define the functional ${\tilde{I}}_{\lambda }^{\prime }:X\to X^{*}$ as

$$\begin{array}{cc}\hfill 〈{\tilde{I}}_{\lambda }^{\prime }\left(v\right),u〉=& {\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{\alpha }_0{|v\left(x\right)-v\left(z\right)|}^{p-2}(v\left(x\right)-v\left(z\right))(u\left(x\right)-u\left(z\right))\mathcal{K}(y,z)dydz\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|v|}^{p-2}vudx-\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{{|v|}^{p-2}v}{{|x|}^{sp}}}udx-{\int }_{{ℝ}^N}g(x,u)vdx\hfill \end{array}$$

for any $u,v\in \mathfrak{E}\left({ℝ}^N\right)$. Since ${\varphi }_n\rightharpoonup \varphi$ in $\mathfrak{E}\left({ℝ}^N\right)$ as $n\to \infty$,

$$〈{\tilde{I}}_{\lambda }^{\prime }\left(\varphi \right),{\varphi }_n-\varphi 〉\to 0\mathrm{as}n\to \infty .$$

(28)

Furthermore, using (19), (20), and the Brézis and Lieb lemma in [52] (Theorem 1), we obtain

$$\begin{array}{cc}\hfill {\left[{\varphi }_n\right]}_{p,\mathcal{K}}^p& -{[{\varphi }_n-\varphi ]}_{p,\mathcal{K}}^p={\left[{\varphi }_n\right]}_{p,\mathcal{K}}^p+o\left(1\right),\hfill \\ \hfill \Vert {\varphi }_n{\Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p& -\Vert {\varphi }_n-{\varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p={\Vert \varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p+o\left(1\right)\hfill \end{array}$$

(29)

and

$$\Vert {\varphi }_n{\Vert }_{H_p}^p-\Vert {\varphi }_n-{\varphi \Vert }_{H_p}^p={\Vert \varphi \Vert }_{H_p}^p+o\left(1\right)$$

(30)

as $n\to \infty$. This, together with (3), (23)–(30) yields that

$$\begin{array}{cc}\hfill o\left(1\right)& =〈I_{\lambda }^{\prime }\left({\varphi }_n\right)-I_{\lambda }^{\prime }\left(\varphi \right),{\varphi }_n-\varphi 〉\hfill \\ \hfill & =M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\tilde{\mathrm{\Psi }}}_{{\varphi }_n}({\varphi }_n-\varphi )-M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right){\tilde{\mathrm{\Psi }}}_{\varphi }({\varphi }_n-\varphi )\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)\left(|{\varphi }_n{|}^{p-2}{\varphi }_n-{|\varphi |}^{p-2}\varphi \right)({\varphi }_n-\varphi )dx\hfill \\ \hfill & -\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^{p-2}{\varphi }_n}{{|x|}^{sp}}}({\varphi }_n-\varphi )dx+\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}({\varphi }_n-\varphi )dx\hfill \\ \hfill & -{\int }_{{ℝ}^N}\left(g(x,{\varphi }_n)-g(x,\varphi )\right)({\varphi }_n-\varphi )dx\hfill \\ \hfill & =M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)\left({\tilde{\mathrm{\Psi }}}_{{\varphi }_n}({\varphi }_n-\varphi )-{\tilde{\mathrm{\Psi }}}_{\varphi }({\varphi }_n-\varphi )\right)\hfill \\ \hfill & +\left(M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)-M\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)\right){\tilde{\mathrm{\Psi }}}_{\varphi }({\varphi }_n-\varphi )\hfill \\ \hfill & +{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right)\left(|{\varphi }_n{|}^{p-2}{\varphi }_n-{|\varphi |}^{p-2}\varphi \right)({\varphi }_n-\varphi )dx\hfill \\ \hfill & -\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^{p-2}{\varphi }_n}{{|x|}^{sp}}}({\varphi }_n-\varphi )dx+\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}({\varphi }_n-\varphi )dx\hfill \\ \hfill & -{\int }_{{ℝ}^N}\left(g(x,{\varphi }_n)-g(x,\varphi )\right)({\varphi }_n-\varphi )dx\hfill \\ \hfill & \ge min\{{\alpha }_0,1\}\left({\tilde{\mathrm{\Psi }}}_{{\varphi }_n}({\varphi }_n-\varphi )+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p-2}{\varphi }_n({\varphi }_n-\varphi )dx\right)\hfill \\ \hfill & -\lambda {\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^{p-2}{\varphi }_n}{{|x|}^{sp}}}({\varphi }_n-\varphi )dx-{\int }_{{ℝ}^N}g(x,{\varphi }_n)({\varphi }_n-\varphi )dx-〈{\tilde{I}}_{\lambda }^{\prime }\left(\varphi \right),{\varphi }_n-\varphi 〉\hfill \\ \hfill & \ge min\{{\alpha }_0,1\}({\int }_{{ℝ}^N}{\int }_{{ℝ}^N}|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right){|}^p\mathcal{K}(x,z)dxdz+{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{|{\varphi }_n\left(x\right)-{\varphi }_n\left(z\right)|}^{p-2}({\varphi }_n\left(x\right)-{\varphi }_n\left(z\right))(\varphi \left(x\right)-\varphi \left(z\right))\mathcal{K}(x,z)dxdz\hfill \\ \hfill & -{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p-2}{\varphi }_n\varphi dx)-\lambda \left({\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx\right)+o\left(1\right)\hfill \\ \hfill & \ge min\{{\alpha }_0,1\}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}^p+\Vert {\varphi }_n{\Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p-{\left[\varphi \right]}_{p,\mathcal{K}}^p-{\Vert \varphi \Vert }_{L^p(\mathfrak{a},{ℝ}^N)}^p\right)\hfill \\ \hfill & -\lambda \left({\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx\right)+o\left(1\right)\hfill \\ \hfill & \ge min\{{\alpha }_0,1\}\left(\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p\right)-\lambda (\Vert {\varphi }_n{\Vert }_{H_p}^p-{\Vert \varphi \Vert }_{H_p}^p)+o\left(1\right)\hfill \\ \hfill & \ge min\{{\alpha }_0,1\}\left(\Vert {\varphi }_n-\varphi {\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p\right)-\lambda (\Vert {\varphi }_n-{\varphi \Vert }_{H_p}^p)+o\left(1\right)\hfill \end{array}$$

(31)

as $n\to \infty$. Thus, we arrive by (19) that

$$\begin{array}{cc}\hfill min\{{\alpha }_0,1\}\Vert {\varphi }_n-\varphi {\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p& \le \lambda \Vert {\varphi }_n-\varphi {\Vert }_{H_p}^p+o\left(1\right)\hfill \\ \hfill & =\lambda \ell +o\left(1\right)\hfill \end{array}$$

(32)

as $n\to \infty$. Now, suppose for contradiction that $\ell >0$. Then, due to Lemma 4, (32) and the fact that $\lambda <min\{{\alpha }_0,1\}{\gamma }_0{\mathcal{C}}_{\mathcal{H}}^{-1}$, we have

$$\begin{array}{cc}\hfill min\{{\alpha }_0,1\}\underset{n\to \infty }{lim}{[{\varphi }_n-\varphi ]}_{p,\mathcal{K}}^p& \le min\{{\alpha }_0,1\}\underset{n\to \infty }{lim}\Vert {\varphi }_n-\varphi {\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p\hfill \\ \hfill & \le \lambda \underset{n\to \infty }{lim}\Vert {\varphi }_n-\varphi {\Vert }_{H_p}^p\hfill \\ \hfill & <min\{{\alpha }_0,1\}{\gamma }_0{\mathcal{C}}_{\mathcal{H}}^{-1}\underset{n\to \infty }{lim}\Vert {\varphi }_n-\varphi {\Vert }_{H_p}^p\hfill \\ \hfill & \le min\{{\alpha }_0,1\}\underset{n\to \infty }{lim}{[{\varphi }_n-\varphi ]}_{p,\mathcal{K}}^p\hfill \end{array}$$

(33)

which is absurd. Consequently, $\ell =0$, so that, by (32), we arrive at ${\varphi }_n\to \varphi$ in $\mathfrak{E}\left({ℝ}^N\right)$. This concludes the proof. □

We are in a position to obtain our first main result. With the aid of Lemmas 5 and 7, we establish the following existence result that (1) admits a sequence of infinitely many large energy solutions.

Proof of Theorem 1. 

Immediately, $I_{\lambda }$ is an even functional and ensures the ${\left(C\right)}_c$-condition for $c\in ℝ$ by Lemma 9. It suffices to verify that there exist ${\beta }_k>{\alpha }_k>0$ such that

(1)

$b_k:=inf\{I_{\lambda }{\left(\varphi \right):\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\alpha }_k,\varphi \in {\mathfrak{Z}}_k\}\to \infty \mathrm{as}k\to \infty$;

(2)

$a_k:=max\{I_{\lambda }{\left(\varphi \right):\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\beta }_k,\varphi \in {\mathfrak{Y}}_k\}\le 0,$

for k large enough. For convenience, we denote

$${\nu }_{1,k}=\underset{{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}=1,\varphi \in {\mathfrak{Z}}_k}{sup}{\Vert \varphi \Vert }_{L^p\left({\mathbb{R}}^N\right)},{\nu }_{2,k}=\underset{{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}=1,\varphi \in {\mathfrak{Z}}_k}{sup}{\Vert \varphi \Vert }_{L^q\left({\mathbb{R}}^N\right)}.$$

Then, it is easy to ensure that ${\nu }_{1,k}\to 0$ and ${\nu }_{2,k}\to 0$ as $k\to \infty$ (see [44]). Let us denote ${\nu }_k=max\{{\nu }_{1,k},{\nu }_{2,k}\}$. Then, we deduce that ${\nu }_k<1$ for sufficiently large k. For any $\varphi \in {\mathfrak{Z}}_k$, assume that ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}>1$.

First we consider $\lambda \in (0,{\lambda }^{*})$. With the analogous argument to that in (10), it follows from the assumptions (M1)–(M2), ($\mathrm{\Psi }1$), and Lemma 4 that for k large enough

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,\varphi )dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\Vert \varphi \Vert }_{L^{ℝ}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\int }_{{ℝ}^N}{|\varphi |}^{q}dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q.\hfill \end{array}$$

(34)

Choose

$${\alpha }_k={\left({\displaystyle \frac{2p{\nu }_k^q{\sigma }_1}{qmin\left\{\frac{{\alpha }_0}{\vartheta }-\frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0},1\right\}}}\right)}^{{\displaystyle \frac{1}{p-q}}}.$$

Since $p<q$ and ${\nu }_k\to 0$ as $k\to \infty$, we assert ${\alpha }_k\to \infty$ as $k\to \infty$. Hence, if $\varphi \in {\mathfrak{Z}}_k$ and ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\alpha }_k$, then we deduce that

$$I_{\lambda }\left(\varphi \right)\ge {\displaystyle \frac{1}{2p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\alpha }_k^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\alpha }_k\to \infty \mathrm{as}k\to \infty ,$$

which implies (1).

The proof of condition (2) is carried out in a similar fashion to that of Theorem 1.3 of [41]. For the convenience of readers, we give the proof. Suppose that condition (2) does not hold for some k. Then, there exists a sequence $\left\{{\varphi }_n\right\}$ in ${\mathfrak{Y}}_k$ such that

$$\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty \mathrm{as}n\to \infty \mathrm{and}{I}_{\lambda }\left({\varphi }_n\right)\ge 0.$$

(35)

Let ${\phi }_n={\varphi }_n/\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}$. Then, $\Vert {\phi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}=1$. Since $dim{\mathfrak{Y}}_k<\infty$, we can choose $\phi \in {\mathfrak{Y}}_k\setminus \left\{0\right\}$ such that, up to a subsequence,

$$\Vert {\phi }_n-\phi {\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to 0\mathrm{and}{\phi }_n\left(x\right)\to \phi \left(x\right)\mathrm{for}\mathrm{a}.\mathrm{e}.x\in {ℝ}^N\mathrm{as}n\to \infty .$$

We prove that $\phi \left(x\right)=0$ for almost all $x\in {\mathbb{R}}^N$. If $\phi \left(x\right)\ne 0$, then $|{\varphi }_n\left(x\right)|\to \infty$ for almost all $x\in {ℝ}^N$ as $n\to \infty$. In accordance with ($\mathrm{\Psi }5$), it follows

$$\underset{n\to \infty }{lim}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}=\underset{n\to \infty }{lim}{\displaystyle \frac{G(x,{\varphi }_n)}{|{\varphi }_n{|}^{p\vartheta }}}{|{\phi }_n|}^{p\vartheta }=\infty$$

(36)

for all $x\in {\mathrm{\Gamma }}_2:=\left\{x\in {ℝ}^N:\phi \left(x\right)\ne 0\right\}$. Using (36) and the Fatou Lemma, one has

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx& \ge \underset{n\to \infty }{lim\; inf}{\int }_{{\mathrm{\Gamma }}_2}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx\hfill \\ & \ge {\int }_{{\mathrm{\Gamma }}_2}\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx=\infty .\hfill \end{array}$$

Thus, we infer

$${\int }_{{\mathbb{R}}^N}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx\to \infty \mathrm{as}n\to \infty .$$

We may suppose that $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}>1$. Using relation (17), we have, for any $\lambda \in (-\infty ,{\lambda }^{*}),$

$$\begin{array}{cc}\hfill I_{\lambda }\left({\varphi }_n\right)& \le 3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }-{\int }_{{\mathbb{R}}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & =\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }\left(3^{\vartheta }max\left\{\mathcal{M}\left(1\right),-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}-{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{G(x,{\varphi }_n)}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{p\vartheta }}}dx\right)\to -\infty \mathrm{as}n\to \infty ,\hfill \end{array}$$

which contradicts (35). The proof is completed. □

Before delving into Theorem 2, we give that $I_{\lambda }$ satisfies the ${\left(C\right)}_c^{*}$-condition, which is the condition of Lemma 6. The proof of this assertion follows the basic idea of that of papers [20,44].

Lemma 8. 

Suppose that (V), (M1)–(M2), ($\mathrm{\Psi }1$)–($\mathrm{\Psi }2$), and ($\mathrm{\Psi }5$) are satisfied. Then, $I_{\lambda }$ satisfies the ${\left(C\right)}_c^{*}$-condition for every $\lambda \in (0,{\lambda }^{*})$, where ${\lambda }^{*}$ is given in (6).

Proof. 

Let $c\in \mathbb{R}$ and let the sequence $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ be such that $v_n\in {\mathfrak{Y}}_n$, for any $n\in N$,

$$I_{\lambda }\left({\varphi }_n\right)\to cand\Vert {\left(I_{\lambda }{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right){\Vert }_{\mathfrak{E}{\left({ℝ}^N\right)}^{*}}(1+\Vert {\varphi }_n\Vert )\to 0\mathrm{as}n\to \infty .$$

Therefore, we obtain $c=I_{\lambda }\left({\varphi }_n\right)+o_n\left(1\right)$ and $〈I_{\lambda }^{\prime }\left({\varphi }_n\right),{\varphi }_n〉=o_n\left(1\right),$ where $o_n\left(1\right)\to 0$ as $n\to \infty$. Repeating the argument as in the proof of Step 1 of Lemma 7, we assure the boundedness of $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ for any $\lambda \in (-\infty ,{\lambda }^{*})$. So, there are a subsequence, still denoted by ${\left\{{\varphi }_n\right\}}_{n\in \mathbb{N}}$, and a function ${\varphi }_0$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that (19) is satisfied.

To complete this proof, we will prove that ${\varphi }_n\to {\varphi }_0$ in $\mathfrak{E}\left({ℝ}^N\right)$ as $n\to \infty$ and also ${\varphi }_0$ is a critical point of $I_{\lambda }$. As $\mathfrak{E}\left({ℝ}^N\right)=\overline{{\bigcup }_{n\in N}{\mathfrak{Y}}_n}$, for $n\in \mathbb{N}$, we can choose $v_n\in {\mathfrak{Y}}_n$ such that $v_n\to {\varphi }_0$ as $n\to \infty$. Hence, we know we have

$$\langle I_{\lambda }^{\prime }\left({\varphi }_n\right),{\varphi }_n-{\varphi }_0\rangle =\langle {\left(I_{\lambda }{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right),{\varphi }_n-v_n\rangle +\langle {\left(I_{\lambda }{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right),v_n-{\varphi }_0\rangle .$$

Since ${\left(I_{\lambda }{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right)\to 0$, $v_n\to {\varphi }_0$ and ${\varphi }_n-v_n\to 0$ in ${\mathfrak{Y}}_n$ as $k\to \infty$, we have

$$\langle I_{\lambda }^{\prime }\left({\varphi }_n\right),{\varphi }_n-{\varphi }_0\rangle \to 0\mathrm{as}n\to \infty .$$

From the analogous arguments to those in the relations (28)–(33) of Lemma 7, we arrive at ${\varphi }_n\to {\varphi }_0$ as $n\to \infty$. Moreover, we have $I_{\lambda }^{\prime }\left({\varphi }_n\right)\to I_{\lambda }^{\prime }\left({\varphi }_0\right)$ as $n\to \infty$. Let us show that ${\varphi }_0$ is a critical point of $I_{\lambda }$. Indeed, fix $n_0\in \mathbb{N}$ and take any $u\in {\mathfrak{Y}}_{n_0}$. We have, for $n\ge n_0,$

$$\langle I_{\lambda }^{\prime }\left({\varphi }_0\right),u\rangle =\langle I_{\lambda }^{\prime }\left({\varphi }_0\right)-I_{\lambda }^{\prime }\left({\varphi }_n\right),u\rangle +\langle I_{\lambda }^{\prime }\left({\varphi }_n\right),u\rangle =\langle I_{\lambda }^{\prime }\left({\varphi }_0\right)-I_{\lambda }^{\prime }\left({\varphi }_n\right),u\rangle +\langle {\left(I_{\lambda }^{\prime }{|}_{{\mathfrak{Y}}_n}\right)}^{\prime }\left({\varphi }_n\right),u\rangle ,$$

so, taking the limit on the right side of the above equation, as $n\to \infty$, we obtain

$$\langle I_{\lambda }^{\prime }\left({\varphi }_0\right),u\rangle =0\mathrm{for}\mathrm{all}u\in {\mathfrak{Y}}_{n_0}.$$

As $n_0$ is taken arbitrarily and ${\bigcup }_{n\in N}{\mathfrak{Y}}_n$ is dense in $\mathfrak{E}\left({ℝ}^N\right)$, we have $I_{\lambda }^{\prime }\left({\varphi }_0\right)=0$, as claimed. Then, for any $\lambda \in (-\infty ,{\lambda }^{*})$, we derive that $I_{\lambda }$ ensures the ${\left(C\right)}_c^{*}$-condition for any $c\in ℝ$. □

Proof of Theorem 2. 

Obviously, the functional $I_{\lambda }$ is even. Thanks to Lemma 8, we note that $I_{\lambda }$ verifies the ${\left(C\right)}_c^{*}$-condition for every $c\in [c_{k_0},0)$. Now, we ensure the properties ($D_1$), ($D_2$), and ($D_3$) in Lemma 6. To carry this out, let ${\nu }_k<1$ for k large enough, where ${\nu }_k$ is given in Theorem 3.

(D1): First, let us consider $\lambda \in (0,{\lambda }^{*})$. From ($\mathrm{\Psi }1$), the definition of ${\nu }_k$, and the analogous arguments to those in (10), it follows that

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,\varphi )dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\Vert \varphi \Vert }_{L^r\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\int }_{{ℝ}^N}{|\varphi |}^{q}dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \\ \hfill & \ge {\displaystyle \frac{C_{1,\min }}{p}}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \end{array}$$

for sufficiently large k and ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\ge 1$, where $C_{1,\min }:=min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}$. Choose

$${\beta }_{1,k}={\left({\displaystyle \frac{2p{\nu }_k^q{\sigma }_1}{q{C}_{1,\min }}}\right)}^{{\displaystyle \frac{1}{p-2q}}}.$$

(37)

Let $\varphi \in {\mathfrak{Z}}_k$ with ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\beta }_{1,k}>1$ for k large enough. Then, we choose a $k_0\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& \ge {\displaystyle \frac{C_{1,\min }}{p}}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{2q}\hfill \\ \hfill & \ge {\displaystyle \frac{C_{1,\min }}{2p}}{\beta }_{1,k}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\beta }_{1,k}\hfill \\ \hfill & \ge 0\hfill \end{array}$$

for all $k\in \mathbb{N}$ with $k\ge k_0$, since ${lim}_{k\to \infty }{\beta }_{1,k}=\infty .$

On the other hand, if $\lambda \in (-\infty ,0]$ and ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\ge 1$, then it follows from ($\mathrm{\Psi }1$), the definition of ${\nu }_k$, and the analogous arguments to those in (12) that

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& \ge {\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx-{\int }_{{ℝ}^N}G(x,\varphi )dx\hfill \\ \hfill & \ge {\displaystyle \frac{C_{2,\min }}{p}}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \end{array}$$

for sufficiently large k, where $C_{2,\min }:=min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }},1\right\}$. Choose

$${\beta }_{2,k}={\left({\displaystyle \frac{2p{\nu }_k^q{\sigma }_1}{q{C}_{2,\min }}}\right)}^{{\displaystyle \frac{1}{p-2q}}}.$$

(38)

Then, we know ${lim}_{k\to \infty }{\beta }_{2,k}=\infty$. Let $\varphi \in {\mathfrak{Z}}_k$ with ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\beta }_{2,k}>1$ for sufficiently large k. Then, we choose a $k_0\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& \ge {\displaystyle \frac{C_{2,\min }}{p}}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{2q}\hfill \\ \hfill & \ge {\displaystyle \frac{C_{2,\min }}{2p}}{\beta }_{2,k}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({ℝ}^N\right)}{\nu }_k{\beta }_{2,k}\hfill \\ \hfill & \ge 0\hfill \end{array}$$

for all $k\in \mathbb{N}$ with $k\ge k_0$.

Let ${\beta }_k$ be either ${\beta }_{1,k}$ or ${\beta }_{2,k}$. Then, we conclude

$$inf\{I_{\lambda }\left(\varphi \right):\varphi \in {\mathfrak{Z}}_k,\Vert \varphi {\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\beta }_k\}\ge 0$$

for any $\lambda \in (-\infty ,{\lambda }^{*})$.

(D2): All the norms are equivalent since ${\mathfrak{Y}}_k$ is finite dimensional. Then, we find positive constants ${\varsigma }_{1,k}$ and ${\varsigma }_{2,k}$ such that

$${\varsigma }_{1,k}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\le {\Vert \varphi \Vert }_{L^{\kappa }(\eta ,{\mathbb{R}}^N)}{\mathrm{and}\Vert \varphi \Vert }_{L^q\left({\mathbb{R}}^N\right)}\le {\varsigma }_{2,k}{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}$$

for any $\varphi \in {\mathfrak{Y}}_k$. From ($\mathrm{\Psi }1$) and ($\mathrm{\Psi }4$), there are ${\mathcal{C}}_1,{\mathcal{C}}_2>0$ such that

$$G(x,\zeta )\ge {\mathcal{C}}_1{\eta \left(x\right)|\zeta |}^{\kappa }-{\mathcal{C}}_2{|\zeta |}^q$$

for almost all $(x,\zeta )\in {\mathbb{R}}^N\times \mathbb{R}$. Let $\varphi \in {\mathfrak{Y}}_k$ with ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\le 1$. First, we consider $\lambda \in (0,{\lambda }^{*})$. Then, we have

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx-{\int }_{{\mathbb{R}}^N}G(x,\varphi )dx\hfill \\ \hfill & \le \left(\underset{0\le \xi \le {\mathcal{C}}_3}{sup}M\left(\xi \right)\right){\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dx+{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx\hfill \\ & -{\mathcal{C}}_1{\int }_{{\mathbb{R}}^N}{\eta \left(x\right)|\varphi |}^{\kappa }dx+{\mathcal{C}}_2{\int }_{{\mathbb{R}}^N}{|\varphi |}^{q}dx\hfill \\ \hfill & \le {\tilde{C}}_1{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\mathcal{C}}_1{\Vert \varphi \Vert }_{L^{\kappa }(\eta ,{ℝ}^N)}^{\kappa }+{\mathcal{C}}_2{\Vert \varphi \Vert }_{L^q\left({ℝ}^N\right)}^q\hfill \\ \hfill & \le {\tilde{C}}_1{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\mathcal{C}}_1{\varsigma }_{1,k}^{\kappa }{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{\kappa }+{\mathcal{C}}_2{\varsigma }_{2,k}^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \end{array}$$

for some positive constants ${\mathcal{C}}_3,{\tilde{C}}_1$.

Next, if $\lambda \in (-\infty ,0)$, we have

$$\begin{array}{cc}\hfill I_{\lambda }\left(\varphi \right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\varphi |}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\varphi |}^p}{{|x|}^{sp}}}dx-{\int }_{{\mathbb{R}}^N}G(x,\varphi )dx\hfill \\ \hfill & \le {\displaystyle \frac{1}{p}}\left(\underset{0\le \xi \le {\mathcal{C}}_3}{sup}M\left(\xi \right)-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}}\right){\int }_{{ℝ}^N}{|\varphi \left(x\right)-\varphi \left(z\right)|}^p\mathcal{K}(x,z)dxdz\hfill \\ & +{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^N}{\mathfrak{a}\left(x\right)|\varphi |}^{p}dx-{\mathcal{C}}_1{\int }_{{\mathbb{R}}^N}{\eta \left(x\right)|\varphi |}^{\kappa }dx+{\mathcal{C}}_2{\int }_{{\mathbb{R}}^N}{|\varphi |}^{q}dx\hfill \\ \hfill & \le {\tilde{C}}_2{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\mathcal{C}}_1{\Vert \varphi \Vert }_{L^{\kappa }(\eta ,{ℝ}^N)}^{\kappa }+{\mathcal{C}}_2{\Vert \varphi \Vert }_{L^q\left({ℝ}^N\right)}^q\hfill \\ \hfill & \le {\tilde{C}}_2{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\mathcal{C}}_1{\varsigma }_{1,k}^{\kappa }{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^{\kappa }+{\mathcal{C}}_2{\varsigma }_{2,k}^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \end{array}$$

for positive constant ${\tilde{C}}_2$.

Let $f\left(\tau \right)={\tilde{C}}_3{\tau }^p-{\mathcal{C}}_1{\varsigma }_{1,k}^{\kappa }{\tau }^{\kappa }+{\mathcal{C}}_2{\varsigma }_{2,k}^q{\tau }^q$ where ${\tilde{C}}_3$ is either ${\tilde{C}}_1$ or ${\tilde{C}}_2$. Since $\kappa <p<q$, one has $f\left(\tau \right)<0$ for all $\tau \in (0,{\tau }_0)$ and for sufficiently small ${\tau }_0\in (0,1)$. Thus, we can choose an ${\alpha }_k>0$ such that $I_{\lambda }\left(\varphi \right)<0$ for all $\varphi \in {\mathfrak{Y}}_k$ with ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\alpha }_k<{\tau }_0$ for sufficiently large k. If necessary, we can replace a large value instead of $k_0$, so that ${\beta }_k>{\alpha }_k>0$ and

$$b_k:=max\{I_{\lambda }\left(\varphi \right):\varphi \in {\mathfrak{Y}}_k{,\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\alpha }_k\}<0,$$

where ${\beta }_k$ is given in (D1).

(D3): Because ${\mathfrak{Y}}_k\cap {\mathfrak{Z}}_k\ne \varphi$ and $0<{\alpha }_k<{\beta }_k$, we have $c_k\le b_k<0$ for all $k\ge k_0$. Let $\varphi \in {\mathfrak{Z}}_k$ with ${\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}=1$ and $0<\tau <{\beta }_k$. With a similar argument to that in (10), we have for any $\lambda \in (0,{\lambda }^{*})$

$$\begin{array}{cc}\hfill I_{\lambda }\left(\tau \varphi \right)& ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\tau \varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\tau \varphi |}^{p}dx-{\displaystyle \frac{\lambda }{p}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\tau \varphi |}^p}{{|x|}^{sp}}}dx-{\int }_{{\mathbb{R}}^N}G(x,\tau \varphi )dx\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}min\left\{{\displaystyle \frac{{\alpha }_0}{\vartheta }}-{\displaystyle \frac{\lambda {\mathcal{C}}_{\mathcal{H}}}{{\gamma }_0}},1\right\}{\Vert \tau \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}{\Vert \tau \varphi \Vert }_{L^r\left({\mathbb{R}}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\Vert \tau \varphi \Vert }_{L^q\left({\mathbb{R}}^N\right)}^q\hfill \\ \hfill & \ge -\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}\tau {\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\tau }^q{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \\ \hfill & \ge -\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}{\beta }_{1,k}{\nu }_k-{\displaystyle \frac{{\sigma }_1}{q}}{\beta }_{1,k}^q{\nu }_k^q\hfill \end{array}$$

for sufficiently large k.

On the other hand, for any $\lambda \in (-\infty ,0]$, it follows upon a similar proceeding to that in (12) that

$$\begin{array}{cc}\hfill I_{\lambda }\left(\tau \varphi \right)& \ge {\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[\tau \varphi \right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|\tau \varphi |}^{p}dx-{\int }_{{\mathbb{R}}^N}G(x,\tau \varphi )dx\hfill \\ \hfill & \ge -\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}\tau {\nu }_k{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}-{\displaystyle \frac{{\sigma }_1}{q}}{\tau }^q{\nu }_k^q{\Vert \varphi \Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^q\hfill \\ \hfill & \ge -\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}{\beta }_{2,k}{\nu }_k-{\displaystyle \frac{{\sigma }_1}{q}}{\beta }_{2,k}^q{\nu }_k^q\hfill \end{array}$$

for k large enough.

Hence, from the definition of ${\beta }_{i,k}$$(i=1,2)$, we infer

$$\begin{array}{cc}\hfill 0>c_k& \ge -\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}{\beta }_{i,k}{\nu }_k-{\displaystyle \frac{{\sigma }_1}{q}}{\beta }_{i,k}^q{\nu }_k^q\hfill \\ \hfill & =-\Vert {\sigma }_0{\Vert }_{L^{r^{\prime }}\left({\mathbb{R}}^N\right)}{\left[{\displaystyle \frac{2p{\sigma }_1}{q{C}_{i,\min }}}\right]}^{{\displaystyle \frac{1}{p-2q}}}{\nu }_k^{{\displaystyle \frac{p-q}{p-2q}}}\hfill \\ \hfill & -{\displaystyle \frac{{\sigma }_1}{q}}{\left[{\displaystyle \frac{2p{\sigma }_1}{q{C}_{i,\min }}}\right]}^{{\displaystyle \frac{q}{p-2q}}}{\nu }_k^{{\displaystyle \frac{q(p-q)}{p-2q}}}.\hfill \end{array}$$

(39)

Because $p<q$ and ${\nu }_k\to 0$ as $k\to \infty$, we conclude that ${lim}_{k\to \infty }{c}_k=0$.

Therefore, all properties of Lemma 6 are fulfilled, and we assert that problem (1) admits a sequence of nontrivial solutions $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$ such that $I_{\lambda }\left({\varphi }_n\right)\to 0$ as $n\to \infty$ for all $\lambda \in (-\infty ,{\lambda }^{*})$. □

4. Proof of Theorems 3 and 4

Next, we will show the ${\left(C\right)}_c$-condition when g satisfies ($\mathrm{\Psi }6$) instead of ($\mathrm{\Psi }5$). However, if $\lambda \in (0,{\lambda }^{*})$ in Lemma 7, then we cannot obtain this result. Hence, we obtain the following assumption when $\lambda \in (-\infty ,0)$. For the convenience of the readers, we set $\tilde{\lambda }=-\lambda$.

Lemma 9. 

Let (V),(M1)–(M3), and ($\mathrm{\Psi }1$)–($\mathrm{\Psi }2$) and ($\mathrm{\Psi }6$) hold. Then, the functional $I_{\tilde{\lambda }}$ ensures the ${\left(C\right)}_c$-condition for any $\tilde{\lambda }>0$.

Proof. 

For any $c\in \mathbb{R}$, let $\left\{{\varphi }_n\right\}$ be a ${\left(C\right)}_c$-sequence in $\mathfrak{E}\left({ℝ}^N\right)$ satisfying (3). In view of Lemma 7, it suffices to verify the boundedness of the sequence $\left\{{\varphi }_n\right\}$ in $\mathfrak{E}\left({ℝ}^N\right)$, using a contradiction for proving the fact that $\left\{{\varphi }_n\right\}$ is bounded in $\mathfrak{E}\left({ℝ}^N\right)$. We suppose that $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}>1$ and $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty$ as $n\to \infty$, and define a sequence $\left\{{\phi }_n\right\}$ by ${\phi }_n={\varphi }_n/\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}$. Then, up to a subsequence, still denoted by $\left\{{\phi }_n\right\}$, we infer ${\phi }_n\rightharpoonup \phi$ in $\mathfrak{E}\left({ℝ}^N\right)$ as $n\to \infty$ and satisfies (4). From the same argument as that in Lemma 7, we have that $\phi \left(x\right)=0$ for almost all $x\in {ℝ}^N$. Since $I_{\tilde{\lambda }}\left(\sigma {\varphi }_n\right)$ is continuous in $\sigma \in [0,1]$, for each $n\in \mathbb{N}$, there exists a ${\sigma }_n$ in $[0,1]$ such that

$$I_{\tilde{\lambda }}\left({\sigma }_n{\varphi }_n\right):=\underset{\sigma \in [0,1]}{max}{I}_{\tilde{\lambda }}\left(\sigma {\varphi }_n\right).$$

Let $\left\{{\varrho }_k\right\}$ be a positive sequence of real numbers fulfilling ${lim}_{k\to \infty }{\varrho }_k=\infty$ and ${\varrho }_k>1$ for any k. Then, $\Vert {\varrho }_k{\phi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}={\varrho }_k>1$ for any k and n. Fix k; since ${\phi }_n\to 0$ strongly in the spaces $L^q\left({ℝ}^N\right)$ as $n\to \infty$, we obtain by the continuity of Nemytskii operator that $G(x,{\varrho }_k{\phi }_n)\to 0$ in $L^1\left({ℝ}^N\right)$ as $n\to \infty$. Hence, we assert

$$\underset{n\to \infty }{lim}{\int }_{{ℝ}^N}G(x,{\varrho }_k{\phi }_n)dx=0.$$

(40)

Since $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}\to \infty$ as $n\to \infty$, we have $\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}>{\varrho }_k$ for large enough n. Hence, we obtain by (40) that

$$\begin{array}{cc}\hfill I_{\tilde{\lambda }}\left({\sigma }_n{\varphi }_n\right)& \ge I_{\tilde{\lambda }}\left({\displaystyle \frac{{\varrho }_k}{\Vert {\varphi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}}}{\varphi }_n\right)=I_{\tilde{\lambda }}\left({\varrho }_k{\phi }_n\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varrho }_k{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varrho }_k{\phi }_n|}^{p}dx\hfill \\ \hfill & +{\displaystyle \frac{\tilde{\lambda }}{p}}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\varrho }_k{\phi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,{\varrho }_k{\phi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{{\alpha }_0}{p\vartheta }}{\int }_{{ℝ}^N}|{\varrho }_k{\phi }_n\left(x\right)-{\varrho }_k{\phi }_n\left(z\right){|}^p\mathcal{K}(x,z)dxdz\hfill \\ \hfill & +{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varrho }_k{\phi }_n|}^{p}dx-{\int }_{{ℝ}^N}G(x,{\varrho }_k{\phi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{min\left\{{\alpha }_0,\vartheta \right\}}{\vartheta p}}\Vert {\varrho }_k{\phi }_n{\Vert }_{\mathfrak{E}\left({ℝ}^N\right)}^p-{\int }_{{ℝ}^N}G(x,{\varrho }_k{\phi }_n)dx\hfill \\ \hfill & \ge {\displaystyle \frac{min\left\{{\alpha }_0,\vartheta \right\}}{\vartheta p}}{\varrho }_k^p\hfill \end{array}$$

for n large enough. Then, letting n and k approach infinity, it follows that

$$\underset{n\to \infty }{lim}{I}_{\tilde{\lambda }}\left({\sigma }_n{\varphi }_n\right)=\infty .$$

(41)

Since $I_{\tilde{\lambda }}\left(0\right)=0$ and $I_{\tilde{\lambda }}\left({\varphi }_n\right)\to c$ as $n\to \infty$, ${\sigma }_n\in (0,1)$, and $〈I_{\tilde{\lambda }}^{\prime }\left({\sigma }_n{\varphi }_n\right),{\sigma }_n{\varphi }_n〉=0$, as a consequence, in accordance with the assumption ($\mathrm{\Psi }6$), we infer that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mu }}{I}_{\tilde{\lambda }}\left({\sigma }_n{\varphi }_n\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\mu }}{I}_{\tilde{\lambda }}\left({\sigma }_n{\varphi }_n\right)-{\displaystyle \frac{1}{p\vartheta \mu }}〈I_{\tilde{\lambda }}^{\prime }\left({\sigma }_n{\varphi }_n\right),{\sigma }_n{\varphi }_n〉\hfill \\ \hfill & ={\displaystyle \frac{1}{p\mu }}\mathcal{M}\left({\left[{\sigma }_n{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\displaystyle \frac{1}{p\mu }}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\sigma }_n{\varphi }_n|}^{p}dx+{\displaystyle \frac{\tilde{\lambda }}{p\mu }}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\sigma }_n{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & -{\displaystyle \frac{1}{\mu }}{\int }_{{ℝ}^N}G(x,{\sigma }_n{\varphi }_n)dx-{\displaystyle \frac{1}{p\vartheta \mu }}M\left({\left[{\sigma }_n{\varphi }_n\right]}_{p,\mathcal{K}}\right){\left[{\sigma }_n{\varphi }_n\right]}_{p,\mathcal{K}}\hfill \\ \hfill & -{\displaystyle \frac{1}{p\vartheta \mu }}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\sigma }_n{\varphi }_n|}^{p}dx-{\displaystyle \frac{\tilde{\lambda }}{p\vartheta \mu }}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\sigma }_n{\varphi }_n{|}^p}{{|x|}^{sp}}}dx+{\displaystyle \frac{1}{p\vartheta \mu }}{\int }_{{ℝ}^N}g(x,{\sigma }_n{\varphi }_n){\sigma }_n{\varphi }_{n}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{p\vartheta \mu }}\left[\vartheta \mathcal{M}\left({\sigma }_n^p{\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)-M\left({\sigma }_n^p{\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)\left({\sigma }_n^p{\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)\right]\hfill \\ \hfill & +{\sigma }_n^p\left({\displaystyle \frac{1}{p\mu }}-{\displaystyle \frac{1}{p\vartheta \mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+\tilde{\lambda }\left({\displaystyle \frac{1}{p\mu }}-{\displaystyle \frac{1}{p\vartheta \mu }}\right){\int }_{{ℝ}^N}{\displaystyle \frac{|{\sigma }_n{\varphi }_n{|}^p}{{|x|}^{sp}}}dx\hfill \\ \hfill & +{\displaystyle \frac{1}{p\vartheta \mu }}{\int }_{{ℝ}^N}\left(g(x,{\sigma }_n{\varphi }_n)\left({\sigma }_n{\varphi }_n\right)-p\vartheta G(x,{\sigma }_n{\varphi }_n)\right)dx\hfill \\ \hfill & ={\displaystyle \frac{1}{p\vartheta \mu }}\widehat{\mathcal{M}}\left({\sigma }_n^p{\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+{\sigma }_n^p\left({\displaystyle \frac{1}{p\mu }}-{\displaystyle \frac{1}{p\vartheta \mu }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & +\tilde{\lambda }\left({\displaystyle \frac{1}{p\mu }}-{\displaystyle \frac{1}{p\vartheta \mu }}\right){\int }_{{ℝ}^N}{\displaystyle \frac{|{\sigma }_n{\varphi }_n{|}^p}{{|x|}^{sp}}}dx+{\displaystyle \frac{1}{p\vartheta \mu }}{\int }_{{ℝ}^N}\mathcal{G}(x,{\sigma }_n{\varphi }_n)dx\hfill \\ \hfill & \le {\displaystyle \frac{1}{p\vartheta }}\widehat{\mathcal{M}}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)+\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{p\vartheta }}\right){\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & +\tilde{\lambda }\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{p\vartheta }}\right){\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx+{\displaystyle \frac{1}{p\vartheta }}{\int }_{{ℝ}^N}\mathcal{G}(x,{\varphi }_n)dx+\tilde{\mathfrak{K}}\hfill \\ \hfill & ={\displaystyle \frac{1}{p}}\mathcal{M}\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right)-{\displaystyle \frac{1}{p\vartheta }}M\left({\left[{\varphi }_n\right]}_{p,\mathcal{K}}\right){\left[{\varphi }_n\right]}_{p,\mathcal{K}}+{\displaystyle \frac{1}{p}}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx\hfill \\ \hfill & -{\displaystyle \frac{1}{p\vartheta }}{\int }_{{ℝ}^N}\mathfrak{a}\left(x\right){|{\varphi }_n|}^{p}dx+{\displaystyle \frac{\tilde{\lambda }}{p}}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\displaystyle \frac{\tilde{\lambda }}{p\vartheta }}{\int }_{{ℝ}^N}{\displaystyle \frac{|{\varphi }_n{|}^p}{{|x|}^{sp}}}dx-{\int }_{{ℝ}^N}G(x,{\varphi }_n)dx\hfill \\ \hfill & +{\displaystyle \frac{1}{p\vartheta }}{\int }_{{ℝ}^N}g(x,{\varphi }_n){\varphi }_{n}dx+\tilde{\mathfrak{K}}\hfill \\ \hfill & =I_{\tilde{\lambda }}\left({\varphi }_n\right)-{\displaystyle \frac{1}{p\vartheta }}〈I_{\tilde{\lambda }}^{\prime }\left({\varphi }_n\right),{\varphi }_n〉+\tilde{\mathfrak{K}}\to c+\tilde{\mathfrak{K}}\mathrm{as}n\to \infty ,\hfill \end{array}$$

which is inconsistent with (41). □

Proof of Theorem 3. 

Immediately, $I_{\lambda }$ is an even functional. Also, $I_{\lambda }$ ensures the ${\left(C\right)}_c$-condition by Lemma 9. The proof is the same as that in Theorem 1 for $\lambda \in (-\infty ,0]$. □

Proof of Theorem 4. 

Due to Lemma 8, we note that the functional $I_{\lambda }$ is even and fulfils the ${\left(C\right)}_c^{*}$-condition for every $c\in [c_{k_0},0)$. The proof is essentially the same as that in Theorem 2 for $\lambda \in (-\infty ,0]$. □

5. Conclusions

The present paper is devoted to deriving the various multiplicity results of solutions to nonlinear equations of the Kirchhoff–Schrödinger–Hardy type driven by the nonlocal fractional p-Laplacian. In contrast to the papers [11,12,13,14,53], we obtain our main results by exploiting a variant of the fountain theorem and the dual fountain theorem instead of the critical point theorems as mentioned in the introduction. The novelty of the present paper is to provide our second main consequences, Theorems 3 and 4, when we do not assume the monotonicity of $\widehat{\mathcal{M}}$ which plays an effective role in ensuring the compactness condition of the Palais–Smale type. In addition, more complicated analyses than those of the previous related works must be performed meticulously because of the presence of a nonlocal Kirchhoff coefficient M and Hardy potential.

In addition, a new research direction in the strong relation is to investigate mixed local and nonlocal problems with the Hardy potential as follows:

$$-M\left({\left[\varphi \right]}_{s,p}\right)\left({∆}_p\varphi +{\mathcal{L}}_p^s\varphi \right)+\mathfrak{a}\left(x\right){|\varphi |}^{p-2}\varphi =\lambda {\displaystyle \frac{{|\varphi |}^{p-2}\varphi }{{|x|}^{sp}}}+g(x,\varphi )\mathrm{in}{ℝ}^N,$$

where ${∆}_p\varphi ={\mathrm{div}(|\nabla \varphi |}^{p-2}\nabla \varphi )$ is the classical p-Laplacian operator. As far as we are aware, there are no consequences for the existence of solutions to the mixed local and nonlocal problems of the Kirchhoff–Schrödinger type with Hardy potential.