Infinitely Many Small Energy Solutions to the Double Phase Anisotropic Variational Problems Involving Variable Exponent

Abstract

This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which is different from the previous related works, is to discuss the multiplicity result of non-trivial solutions by applying the dual fountain theorem as the main tool. In particular, our main result is obtained without assuming the conditions on the nonlinear term at infinity.

1. Introduction

The studies of differential equations and variational problems with nonhomogeneous operators and non-standard growth conditions have attracted extensive attention during the last decades. Let us recall some related results by way of motivation. Azzollini et al. in [1,2] introduced a new class of nonhomogeneous operators with a variational structure:

$$-div({\phi }^{\prime }{\left(\right|\nabla w|}^2)\nabla w),$$

where $\phi \in C^1({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ has a different growth near zero and infinity. As noted in [1], the theory of classical Sobolev spaces is not applicable, because the different growth of the principal part and an unbounded domain are considered. Hence, the study of these nonhomogeneous differential operators is based on the theory of Orlicz–Sobolev spaces. In this reason, in order to obtain the existence results, the authors of [1] provided an adequate functional framework based on the paper [3]. This functional setting is considered in the sum of Lebesgue spaces which can be regarded as an Orlicz–Sobolev space. In particular, Azzollini, d’Avenia and Pomponio in [1] established the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle -div({\phi }^{\prime }{\left(\right|\nabla w|}^2)\nabla w)+{\left|w\right|}^{\alpha -2}w={\left|w\right|}^{\ell -2}w\mathrm{in}{\mathbb{R}}^N,}\hfill \\ w=0,as\left|x\right|\to \infty ,\hfill \end{array}\right.\end{array}$$

where $N\ge 2$, $\phi \left(t\right)$ behaves like $t^{s/2}$ for small t and $t^{r/2}$ for large t, and

$$1<r<s<N,1<\alpha \le \frac{r^{*}{s}^{\prime }}{r^{\prime }},max\{\alpha ,s\}<\ell <r^{*},$$

(1)

with $r^{\prime }=r/(r-1)$ and $s^{\prime }=s/(s-1)$. Under the above condition (1), Chorfi and Rădulescu [4] obtained the existence of the standing wave solutions to the following Schrödinger equation with unbounded potential:

$$-div({\phi }^{\prime }{\left(\right|\nabla w|}^2)\nabla w)+{a\left(x\right)\left|w\right|}^{\alpha -2}w=f(x,w)\mathrm{in}{\mathbb{R}}^N,$$

where the nonlinearity $f:{\mathbb{R}}^N\to \mathbb{R}$ also satisfies the subcritical growth. Very recently, Zhang and Rădulescu [5] further extended the results in [4] to the more general variable exponent case

$$-\mathrm{div}\varphi (x,\nabla w)+{V\left(x\right)\left|w\right|}^{\alpha \left(x\right)-2}w=f(x,w)\mathrm{in}{\mathbb{R}}^N,$$

(2)

where the differential operator $\varphi (x,\xi )$ has behaviors like ${\left|\xi \right|}^{s\left(x\right)-2}\xi$ for small $\xi$ and like ${\left|\xi \right|}^{r\left(x\right)-2}\xi$ for large $\xi$, $V:{\mathbb{R}}^N\to {\mathbb{R}}^{}+$ is a potential function satisfying some conditions, $1<\alpha (·)\le r(·)<s(·)<N$. Motivated by this work, the authors in [6] established the existence of a sequence of large- and small- energy solutions for Schrödinger-type problems involving the double phase operator with concave–convex nonlinearities. The strategy of proof for these results is to employ the fountain theorem and the dual fountain theorem as the main tools, respectively. Moreover, some existence results for problems of this type involving critical growth are investigated in [7].

Problem Equation (2) also corresponds to double phase anisotropic phenomenon. The double phase problems are described by the following functional

$${\int }_{\mathsf{\Omega }}\Phi (x,\xi )dx$$

with the so-called $(r,s)$-growth conditions:

$${c\left|\xi \right|}^r\le \Phi (x,\xi )\le {C\left(\right|\xi |}^s+1).$$

Double phase functionals have been introduced by Zhikov in the context of Homogenization and Lavrentiev’s phenomenon [8,9]. The $(r,s)$-growth condition was first treated by Marcellini [10,11,12,13] and it has been extensively studied in the last decades. For an overview of the subject, we refer the readers to the survey paper [14]. The study of differential equations and variational problems involving double phase operator has been paid to a great deal of attention in recent years; see [1,5,14,15,16,17,18,19,20,21,22,23]. The interest in variational problems with double phase operator can be corroborated as a model for many physical phenomena which arise in the research of elasticity, plasma physics, strongly anisotropic materials, biophysics and chemical reactions, Lavrentiev’s phenomenon, etc.; see [8,9]. With regard to regularity theory for double phase functionals, we would like to mention a series of notable papers by Mingione et al. [16,17,18,24,25,26]; see also [14,20]. Liu-Dai [22] obtained the various existence and multiplicity results to to nonlinear double phase problems with a constant exponents; see [21] for the variable exponents. Relating to the eigenvalue problem, this can be found in [15,19]. A remarkable inquiry of some of the recent works on two phase equations can be found in Radulescu [23]. Furthermore, we refer to the works of Bahrouni–Rǎdulescu–Repovš [27], Byun–Oh [28], Crespo Blanco–Gasiński-Harjulehto–Winkert [29], Gasiński–Winkert [30], Kim–Kim [31], Papageorgiou–Rǎdulescu–Repovš [32], Perera–Squassina [33], Ragusa–Tachikawa [34] and Zeng–Bai–Gasiński–Winkert [35].

In these respects, the present paper is concerned with the following double phase anisotropic variational problem:

$$-\mathrm{div}\left(\varphi (x,\nabla w)\right)+{V\left(x\right)\left|w\right|}^{\alpha \left(x\right)-2}w=\lambda \kappa \left(x\right){\left|w\right|}^{\gamma \left(x\right)-2}w+f(x,w)\mathrm{in}{\mathbb{R}}^N,$$

(3)

where $\lambda >0$ is a parameter, $\alpha :{\mathbb{R}}^N\to (1,\infty )$ is a function, $\gamma \in C\left({\mathbb{R}}^N\right)$ satisfies $1<\gamma \left(x\right)<r^{*}\left(x\right)$ with $r^{*}\left(x\right):=\frac{Nr\left(x\right)}{N-r\left(x\right)}$ for all $x\in {\mathbb{R}}^N$, V and $\kappa$ are appropriate potential functions, $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function and $\varphi :{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}^N$ admits a potential $\Phi$, i.e., ${\nabla }_{\eta }\Phi (x,\eta )=\varphi (x,\eta )$ for some $\Phi :{\mathbb{R}}^N\times {\mathbb{R}}^N\to \mathbb{R}$.

For t, $\ell \in C\left(R^N\right)$, by $t\ll \ell$ (resp. $t\gg \ell$) we mean ${inf}_{x\in {\mathbb{R}}^N}(\ell \left(x\right)-t\left(x\right))>0$ (resp. ${inf}_{x\in {\mathbb{R}}^N}(t\left(x\right)-\ell \left(x\right))>0$) for almost all $x\in {\mathbb{R}}^N$. We make the following assumptions:

(HA1)

The potential $\Phi =\Phi (x,\eta )$ is a continuous function in ${\mathbb{R}}^N\times {\mathbb{R}}^N$, and has continuous derivative with respect to $\eta$ such that $\varphi ={\partial }_{\eta }\Phi (x,\eta )$.

(HA2)

$\Phi (x,\mathbf{0})=0$ and $\Phi (x,\eta )=\Phi (x,-\eta ),$ for all $(x,\eta )\in {\mathbb{R}}^N\times {\mathbb{R}}^N$.

(HA3)

$\Phi (x,·)$ is strictly convex in ${\mathbb{R}}^N$ for all $x\in {\mathbb{R}}^N$.

(HA4)

$1\ll r\ll s\ll r_N^{*}$ with $r_N^{*}\left(x\right):=min\{N,r^{*}\left(x\right)\}$ for all $x\in {\mathbb{R}}^N$, r and s are Lipschitz continuous in ${\mathbb{R}}^N$.

(HA5)

There exist positive constants $C_1,C_2$ such that for all $(x,\eta )\in {\mathbb{R}}^N\times {\mathbb{R}}^N$

$$\begin{array}{c}\hfill C_1{\left|\eta \right|}^{r\left(x\right)},\mathrm{if}\left|\eta \right|>1\\ \hfill C_1{\left|\eta \right|}^{s\left(x\right)},\mathrm{if}\left|\eta \right|\le 1\end{array}\}\le \varphi (x,\eta )·\eta$$

and

$$\left|\varphi (x,\eta )\right|\le \{\begin{array}{c}{C}_2{\left|\eta \right|}^{r\left(x\right)-1},\mathrm{if}\left|\eta \right|>1\hfill \\ C_2{\left|\eta \right|}^{s\left(x\right)-1},\mathrm{if}\left|\eta \right|\le 1.\hfill \end{array}$$

(HA6)

$\varphi (x,\eta )·\eta \le p\left(x\right)\Phi (x,\eta )$ for any $(x,\eta )\in {\mathbb{R}}^N\times {\mathbb{R}}^N$, where p is Lipschitz continuous and satisfies $s\left(x\right)\le p\left(x\right)$ for all $x\in {\mathbb{R}}^N$ and $p\ll r^{*}$.

(HA7)

The potential $\Phi (x,·)$ is uniformly convex, that is, for any $\epsilon \in (0,1)$ there exists $\delta \left(\epsilon \right)\in (0,1)$ such that $|\eta -\xi |\le \epsilon max\left\{\right|\eta |,|\xi \left|\right\}$ or $\Phi \left(x,\frac{\eta +\xi }{2}\right)\le \frac{1}{2}(1-\delta \left(\epsilon \right))(\Phi (x,\eta )+\Phi (x,\xi ))$ for any $x,\eta ,\xi \in {\mathbb{R}}^N$.

Conditions (HA2) and (HA3) imply that

$$\Phi (x,\eta )\le \varphi (x,\eta )·\eta \mathrm{for}\mathrm{all}(x,\eta )\in {\mathbb{R}}^N\times {\mathbb{R}}^N.$$

Furthermore, (HA3) is weaker than the request that $\Phi$ is uniformly convex, that is, for any $\epsilon \in (0,1)$, there exists a constant $\delta \left(\epsilon \right)\in (0,1)$ such that

$$\Phi \left(x,\frac{\eta +\xi }{2}\right)\le (1-\delta \left(\epsilon \right))\frac{\Phi (x,\eta )+\Phi (x,\xi )}{2}$$

for all $x\in {\mathbb{R}}^N$ and $\eta ,\xi \in {\mathbb{R}}^N\times {\mathbb{R}}^N$ satisfy $|u-v|\ge \epsilon max\left\{\right|\eta |,|\xi \left|\right\}$.

If (HA2), (HA3) and (HA5) hold, then, we have

$$\begin{array}{c}\hfill C_1{\left|\eta \right|}^{r\left(x\right)},\left|\eta \right|>1\\ \hfill C_1{\left|\eta \right|}^{s\left(x\right)},\left|\eta \right|\le 1\end{array}\}\le \Phi (x,\eta )\le \varphi (x,\eta )·\eta \le \{\begin{array}{c}\hfill C_2{\left|\eta \right|}^{r\left(x\right)},\left|\eta \right|>1\\ \hfill C_2{\left|\eta \right|}^{s\left(x\right)},\left|\eta \right|\le 1.\end{array}$$

(4)

Moreover, we assume that $V:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$ satisfies

(V)

$V\in L_{loc}^1\left({\mathbb{R}}^N\right)$, ${\mathrm{essinf}}_{x\in {\mathbb{R}}^N}V\left(x\right)>0$ and $V\left(x\right)\to +\infty$ as $\left|x\right|\to +\infty$.

The purpose of this paper is devoted to the existence result of a sequence of infinitely many small energy solutions to the double phase anisotropic variational problems with concave–convex nonlinearities when the convex term f does not require the Ambrosetti–Rabinowitz condition in [36] as follows, namely, there exists a constant $\theta >0$ such that $\theta >{sup}_{x\in {\mathbb{R}}^N}s\left(x\right)$ and

$$0<\theta F(x,w)\le f(x,w)w\mathrm{for}\mathrm{all}(x,w)\in {\mathbb{R}}^N\times \mathbb{R}\backslash \left\{0\right\},\mathrm{where}F(x,w)={\int }_0^{w}f(x,t)dt.$$

As we know, this condition is crucial to ensure a compactness condition of Palais–Smale type of the Euler–Lagrange functional corresponding to the problem Equation (3). Such multiplicity result is obtained by the dual fountain theorem as the main tool. This result of multiple solutions to nonlinear elliptic problems is motivated by the contributions in recent works [37,38,39,40,41,42,43,44], and the references therein. In particular, the existence and multiplicity results to the superlinear elliptic problems:

$$-{\mathrm{div}\left(\right|\nabla w|}^{r\left(x\right)-2}{\nabla w)+V\left(x\right)|w|}^{r\left(x\right)-2}w=f(x,w)\mathrm{in}{\mathbb{R}}^N.$$

have been investigated in Alves–Liu [39]. Here, the Carathéodory function $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ fulfills the following assumptions:

(f1)

$F(x,w)=o\left(\right|w{|}^{r\left(x\right)})$ as $w\to 0$ uniformly for all $x\in {\mathbb{R}}^N$.

(f2)

${lim}_{\left|w\right|\to \infty }\frac{F(x,w)}{{\left|w\right|}^{r^{+}}}=\infty$ uniformly for almost all $x\in {\mathbb{R}}^N$.

(f3)

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

$$\theta \left(f(x,w)w-r^{+}F(x,w)\right)\ge f(x,tw)tw-r^{+}F(x,tw)$$

for $(x,w)\in {\mathbb{R}}^N\times \mathbb{R}$ and $t\in [0,1]$, where $r^{+}:={sup}_{x\in {\mathbb{R}}^N}r\left(x\right)$.

In the last few decades, there were extensive studies dealing with the r-Laplacian problem by assuming (f3); see [42,43] and see also [44,45] for the case of variable exponents $r(·)$. If we consider the function

$$f(x,\tau )=\rho \left(x\right)\left({\left|\tau \right|}^{\ell \left(x\right)-2}\tau ln\left(1+\left|\tau \right|\right)+\frac{{\left|\tau \right|}^{\ell \left(x\right)-1}\tau }{1+\left|\tau \right|}\right)$$

with its primitive function

$$F(x,\tau )=\frac{\rho \left(x\right)}{\ell \left(x\right)}{\left|\tau \right|}^{\ell \left(x\right)}ln\left(1+\left|\tau \right|\right)$$

for all $\tau \in \mathbb{R}$ and $\ell \in C_{+}\left({\mathbb{R}}^N\right)$, where $s^{+}<\ell \left(x\right)$ for all $x\in {\mathbb{R}}^N$ and $\rho \in C({\mathbb{R}}^N,\mathbb{R})$ with $0<{inf}_{x\in {\mathbb{R}}^N}\rho \left(x\right)\le {sup}_{x\in {\mathbb{R}}^N}\rho \left(x\right)<\infty$. Then, this example fulfills the assumptions (f1)–(f3). On the other hand, let us consider the modified function of the above example as follows:

$$f(x,\tau )=\rho \left(x\right)\left({\eta \left(x\right)\left|\tau \right|}^{{\alpha }_0\left(x\right)-2}\tau +{\left|\tau \right|}^{\ell \left(x\right)-2}\tau ln\left(1+\left|\tau \right|\right)+\frac{{\left|\tau \right|}^{\ell \left(x\right)-1}\tau }{1+\left|\tau \right|}\right)$$

where ${\alpha }_0,\eta$ will be specified later. Then, this example fulfills the assumptions (f2)–(f3), but not (f1).

Recently, Lin–Tang [40] gave the various existence theorems on a sequence of infinitely many solutions to r-Laplacian equations with mild conditions for the superlinear term f which is deeply different from those investigated in [39,42,43,44,45]. Motivated by this work, we give some examples which do not satisfy the assumptions (f1) and (f3).

In this direction, on a class of superlinear term f which is different from the previous related works, we give the existence result of a sequence of infinitely many small energy solutions by utilizing the dual fountain theorem. However, the proof for obtaining this result slightly differs from those of previous related works [6,37,41,46,47]. More precisely, in view of [6,37], the conditions (f1) and (f2) play a decisive role in verifying some assumptions in the dual fountain theorem; however, we ensure them when (f1) and (f2) are not assumed. This is a novelty of the present paper. To the best of our belief, although this work is inspired by the papers [6,38] and many authors have an interest in the investigation of elliptic problems with variable exponents, the present paper is the first attempt to obtain such multiplicity result to the double phase anisotropic variational problems. In particular, the present paper is an improvement of the recent work [6] about the existence of infinitely many small energy solutions because we do not assume the fact $\alpha (·)\le r(·)$ as well as the conditions (f1) and (f2).

This paper is organized as follows. In Section 2, we shortly introduce the definition of the Lebesgue spaces with variable exponents and the fractional variable exponent Lebesgue–Sobolev space, and present some necessary preliminary knowledge of function spaces, which we will use along the paper. Section 3 provides the existence result of infinitely many small energy solutions to the problem Equation (3) by applying the dual fountain theorem as the primary tool.

2. Preliminaries

In order to discuss problem Equation (3), we briefly list some theory of variable exponent Lebesgue–Sobolev spaces and the variable exponent Orlicz–Sobolev space. Afterward, we will give some properties of these variable exponent spaces which were systematically studied in [5,48,49,50,51].

Let

$$C_{+}\left({\mathbb{R}}^N\right)=\left\{r\in C\left({\mathbb{R}}^N\right):\underset{x\in {\mathbb{R}}^N}{inf}r\left(x\right)>1\right\}.$$

For any $r\in C_{+}\left({\mathbb{R}}^N\right),$ let

$$r_{+}=\underset{x\in {\mathbb{R}}^N}{sup}r\left(x\right)\mathrm{and}{r}_{-}=\underset{x\in {\mathbb{R}}^N}{inf}r\left(x\right).$$

For any $p\in C_{+}\left({\mathbb{R}}^N\right)$, the variable exponent Lebesgue space is defined by

$$L^{r(·)}\left({\mathbb{R}}^N\right):=\left\{w:{\mathbb{R}}^N\to \mathbb{R}\mathrm{is}\mathrm{a}\mathrm{measurable}:{\int }_{{\mathbb{R}}^N}{\left|w\left(x\right)\right|}^{r\left(x\right)}dx<\infty \right\}$$

and is endowed with the Luxemburg norm

$${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}=inf\left\{\lambda >0:{\int }_{{\mathbb{R}}^N}|\frac{w\left(x\right)}{\lambda }{|}^{r\left(x\right)}dx\le 1\right\}.$$

The dual space of $L^{r(·)}\left({\mathbb{R}}^N\right)$ is $L^{r^{\prime }(·)}\left({\mathbb{R}}^N\right)$, where $1/r\left(x\right)+1/r^{\prime }\left(x\right)=1$.

The variable exponent Sobolev space $W^{1,r(·)}\left({\mathbb{R}}^N\right)$ is defined by

$$W^{1,r(·)}\left({\mathbb{R}}^N\right):=\left\{w\in L^{r(·)}\left({\mathbb{R}}^N\right):|\nabla w|\in L^{r(·)}\left({\mathbb{R}}^N\right)\right\}$$

with the norm

$${\left|\right|w\left|\right|}_{W^{1,r(·)}\left({\mathbb{R}}^N\right)}={\left|\right|\nabla w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}+{\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}.$$

Lemma 1

([51]). The space $L^{r(·)}\left({\mathbb{R}}^N\right)$ is a separable, uniformly convex Banach space, and its dual space is $L^{r^{\prime }(·)}\left({\mathbb{R}}^N\right)$, where $1/r\left(x\right)+1/r^{\prime }\left(x\right)=1$. For any $w\in L^{r(·)}\left({\mathbb{R}}^N\right)$ and $v\in L^{r^{\prime }(·)}\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{cc}\hfill \left|{\int }_{{\mathbb{R}}^N}wvdx\right|& \le \left(\frac{1}{r_{-}}+\frac{1}{r_{-}^{\prime }}\right){\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}{\left|\right|v\left|\right|}_{L^{r^{\prime }(·)}\left({\mathbb{R}}^N\right)}\hfill \\ \hfill & \le {2\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}{\left|\right|v\left|\right|}_{L^{r^{\prime }(·)}\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Lemma 2

([51]). Let

$$\rho \left(w\right)={\int }_{{\mathbb{R}}^N}{\left|w\right|}^{r\left(x\right)}dx\mathrm{for}\mathrm{all}w\in L^{r(·)}\left({\mathbb{R}}^N\right).$$

Then,

(1)

$\rho \left(w\right)>1(=1;<1)$ if and only if ${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}>1(=1;<1,resp.)$;

(2)

if ${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}>1$, then ${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}^{r_{-}}\le \rho \left(w\right)\le {\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}^{r_{+}}$;

(3)

if ${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}\le 1$, then ${\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}^{r_{+}}\le \rho \left(w\right)\le {\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}^{r_{-}}$.

Lemma 3

([48]). Let $n\in L^{\infty }\left({\mathbb{R}}^N\right)$ be such that $1\le r\left(x\right)s\left(x\right)\le \infty$ for almost all $x\in {\mathbb{R}}^N$. If $w\in L^{r(·)s(·)}\left({\mathbb{R}}^N\right)$ with $w\ne 0$, then

(1)

if ${\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}>1$, then

$${\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}^{s_{-}}\le {\left|\right|\left|w\right|}^{s\left(x\right)}{\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}\le {\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}^{s_{+}};$$

(2)

if ${\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}\le 1$, then

$${\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}^{s_{+}}\le {\left|\right|\left|w\right|}^{s\left(x\right)}{\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}\le {\left|\right|w\left|\right|}_{L^{r(·)s(·)}\left({\mathbb{R}}^N\right)}^{s_{-}}.$$

Lemma 4

([51]). If $w,w_n\in L^{r(·)}\left({\mathbb{R}}^N\right),n=1,2,\cdots ,$ then, the following statements are equivalent:

(1)

${lim}_{n\to \infty }\left|\right|w_n-w{\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}=0;$

(2)

${lim}_{n\to \infty }\rho (w_n-w)=0;$

(3)

$w_n\to w$ for almost every in ${\mathbb{R}}^N$ and ${lim}_{n\to \infty }\rho \left(w_n\right)=\rho \left(w\right)$.

For simplicity, we set ${\left|\right|w\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}:={\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)+L^{s(·)}\left({\mathbb{R}}^N\right)}$.

Definition 1.

Let (HA4) hold. We denote by $L^{r(·)}\left({\mathbb{R}}^N\right)+L^{s(·)}\left({\mathbb{R}}^N\right)$ the completion of $C_c^{\infty }({\mathbb{R}}^N,\mathbb{R})$ in the norm

$$\begin{array}{c}\hfill {\left|\right|u\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}=inf\left\{{\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathbb{R}}^N\right)}+{\left|\right|v\left|\right|}_{L^{s(·)}\left({\mathbb{R}}^N\right)}:w\in L^{r(·)}\left({\mathbb{R}}^N\right),v\in L^{s(·)}\left({\mathbb{R}}^N\right),u=v+w\right\}.\end{array}$$

Proposition 1

([5]). Assume that (HA4) holds. Let Ω be an open domain, $w\in L^{r(·)}\left(\mathsf{\Omega }\right)+L^{s(·)}\left(\mathsf{\Omega }\right)$ and ${\mathsf{\Lambda }}_w=\{x\in \mathsf{\Omega }:|w\left(x\right)|>1\}$. Then, we have

(1)

$|{\mathsf{\Lambda }}_w|<+\infty ;$

(2)

$w\in L^{r(·)}\left({\mathsf{\Lambda }}_w\right)\cap L^{s(·)}\left({\mathsf{\Lambda }}_w^c\right);$

(3)

if $B\subset \mathsf{\Omega },$ then ${\left|\right|w\left|\right|}_{L^{r(·),s(·)}\left(\mathsf{\Omega }\right)}\le {\left|\right|w\left|\right|}_{L^{r(·),s(·)}\left(B\right)}+{\left|\right|w\left|\right|}_{L^{r(·),s(·)}(\mathsf{\Omega }/B)};$

(4)

we have

$$\begin{array}{cc}\hfill & max\left\{\frac{1}{1+2|{\mathsf{\Lambda }}_w{|}^{\frac{1}{r\left(\eta \right)}-\frac{1}{s\left(\eta \right)}}}{\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathsf{\Lambda }}_w\right)},C_{1}min\left\{{\left|\right|w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_w^c\right)}{,\left|\right|w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_w^c\right)}^{\frac{s\left(\eta \right)}{r\left(\eta \right)}}\right\}\right\}\le \hfill \\ \hfill & {\left|\right|w\left|\right|}_{L^{r(·),s(·)}\left(\mathsf{\Omega }\right)}\le {\left|\right|w\left|\right|}_{L^{r(·)}\left({\mathsf{\Lambda }}_w\right)}+{\left|\right|w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_w^c\right)}\le {2max\left\{\right|\left|w\right||}_{L^{r(·)}\left({\mathsf{\Lambda }}_w\right)}{,\left|\right|w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_w^c\right)}\},\hfill \end{array}$$

where $\eta \in {\mathbb{R}}^N$ and c is a small positive constant.

Let ${\alpha }_{-}>1$ and ${\alpha }_{+}<\infty$. Under the condition (V), we denote the space

$$L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)=\left\{w:{\mathbb{R}}^N\to \mathbb{R}\mathrm{is}\mathrm{measurable}:{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\left(x\right)\right|}^{\alpha \left(x\right)}dx<\infty \right\}$$

with the norm

$${\left|\right|w\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}=inf\left\{\lambda >0:{\int }_{{\mathbb{R}}^N}V\left(x\right)|\frac{w\left(x\right)}{\lambda }{|}^{\alpha \left(x\right)}dx\le 1\right\}.$$

Let $\alpha \in C_{+}\left({\mathbb{R}}^N\right)$ and (HA4) hold. Define $X=\left\{w\in L_V^{\alpha (·)}\left({\mathbb{R}}^N\right):\nabla w\in {(L^{r(·)}\left({\mathbb{R}}^N\right)+L^{s(·)}\left({\mathbb{R}}^N\right))}^N\right\}$ with following norm

$${\left|\right|w\left|\right|}_X:={\left|\right|w\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}+{\left|\right|\nabla w\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}.$$

The next following assertions are essential in our study.

Proposition 2

([5]). Let (HA4) hold and (V) hold. Then, the space X with the norm ${\parallel ·\parallel }_X$ is a reflexive Banach space.

Remark 1.

To employ the fountain theorem in the next section, the separability of this space is required. To this end, define

$$\phi (x,t):=min\{t^{r\left(x\right)-1},t^{s\left(x\right)-1}\}$$

for all $t\ge 0$ and $x\in {\mathbb{R}}^N$. Let us consider the Orlicz class

$$L^{\mathsf{{\rm Y}}}\left({\mathbb{R}}^N\right):=\left\{w\in \mathfrak{M}\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}\mathsf{{\rm Y}}(x,w\left(x\right))d\mu \left(x\right)<+\infty \right\},$$

where $\mathfrak{M}\left({\mathbb{R}}^N\right)$ is the linear space of the real measurable functions defined on ${\mathbb{R}}^N$ and

$$\mathsf{{\rm Y}}(x,s):={\int }_0^{\left|s\right|}\phi (x,t)dt$$

for all $s\in \mathbb{R}$ and $x\in {\mathbb{R}}^N$. From an analogous argument in [6], we infer that X is a separable and reflexive Banach space.

Lemma 5

([5]). Assume that (HA4) and (V) hold and let α satisfy $1\ll \alpha (·)\ll r^{*}(·)\frac{N-1}{N}$ and $\alpha (·)\le r^{*}(·)\frac{s^{\prime }(·)}{r^{\prime }(·)}$. Then, the following conclusions hold:

(1)

For any ℓ with $\alpha \left(x\right)\le \ell \left(x\right)\le r^{*}\left(x\right)$ for all $x\in {\mathbb{R}}^N$, there is a continuous embedding $X\hookrightarrow L^{\ell (·)}\left({\mathbb{R}}^N\right);$

(2)

For any bounded subset $\mathsf{\Omega }\subset {\mathbb{R}}^N$, there is a compact embedding $X\left(\mathsf{\Omega }\right)\hookrightarrow \hookrightarrow L^{\ell (·)}\left(\mathsf{\Omega }\right);$

(3)

For any $\ell \in C\left({\mathbb{R}}^N\right)$ which is Lipschitz continuous with $\alpha \left(x\right)\le \ell \left(x\right)\ll r^{*}(·)$ for all $x\in {\mathbb{R}}^N$, there is a compact embedding $X\hookrightarrow \hookrightarrow L^{\ell (·)}\left({\mathbb{R}}^N\right)$.

3. Hypotheses and Main Results

First, we provide some preliminary results about the variational setting of the problem Equation (3), which will be used in obtaining our main result. Then, the existence of nontrivial weak solutions for Equation (3) is provided by applying the dual fountain theorem under suitable assumptions. To do this, assume that

(B1)

$r,s,\gamma \in C_{+}\left({\mathbb{R}}^N\right)$ and $1<\gamma \left(x\right)\ll min\left\{\alpha \right(x),r(x\left)\right\}$ and

$max\{\alpha \left(x\right),r\left(x\right)\}\ll s\left(x\right)\ll min\{N,r^{*}\left(x\right)\}$ for all $x\in {\mathbb{R}}^N$, where $\alpha$ satisfies all conditions in Lemma 5.

(B2)

$0\le \kappa \in L^{\infty }\left({\mathbb{R}}^N\right)\cap L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)$ with $\left|\right\{x\in {\mathbb{R}}^N:\kappa \left(x\right)\ne 0\left\}\right|>0$ for any ${\ell }_0\in C_{+}\left({\mathbb{R}}^N\right)$ with $max\{r\left(x\right),\alpha \left(x\right)\}\le {\ell }_0\left(x\right)\le r^{*}\left(x\right)$ for all $x\in {\mathbb{R}}^N$, where $|·|$ is the Lebesgue measure in ${\mathbb{R}}^N$ and ${\tilde{\ell }}_0\left(x\right):=\frac{{\ell }_0\left(x\right)}{{\ell }_0\left(x\right)-\gamma \left(x\right)}$ for all $x\in {\mathbb{R}}^N$.

(H1)

$f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory condition.

(H2)

There exist $0\le {\sigma }_1\in L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)\cap L^{\infty }\left({\mathbb{R}}^N\right)$ and a positive constant ${\sigma }_2$ such that

$$\left|f(x,t)\right|\le {\sigma }_1\left(x\right)+{\sigma }_2{\left|t\right|}^{\ell \left(x\right)-1}$$

for all $(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$ where ℓ is Lipschitz continuous with $s_{+}<{\ell }_{-}\le {\ell }_{+}<r^{*}\left(x\right)$ for all $x\in {\mathbb{R}}^N$.

(H3)

There exist $C_3>0$, $1<{\alpha }_0\left(x\right)\ll \alpha \left(x\right)$, $\tau \left(x\right)>1$ with $\alpha \left(x\right)\le {\tau }^{\prime }\left(x\right){\alpha }_0\left(x\right)\le r^{*}\left(x\right)$ for all $x\in {\mathbb{R}}^N$ and a positive function $\eta \in L^{\tau (·)}\left({\mathbb{R}}^N\right)\cap L^{\infty }\left({\mathbb{R}}^N\right)$ such that

$$\underset{\left|t\right|\to 0}{lim\; inf}\frac{f(x,t)}{{\eta \left(x\right)\left|t\right|}^{{\alpha }_0\left(x\right)-2}t}\ge C_3$$

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

(H4)

there exist $\nu >s_{+}$, $M>0$ such that

$$\nu F(x,t)\le tf(x,t)$$

for all $x\in {\mathbb{R}}^N$ and $\left|t\right|\ge M$.

We give some simple examples satisfying the conditions (H1)–(H4), but it is clear that they do not satisfy the conditions (f1) and (f3). These examples can be found in [6] (see also [40] for a constant exponent) even if they are slightly different in a sense.

Example 1.

If $1<{\alpha }_0\left(x\right)<\alpha \left(x\right)=r\left(x\right)=2<s\left(x\right)=3$ for all $x\in {\mathbb{R}}^N$ and we consider the function

$$f(x,t)={\eta \left(x\right)\left|t\right|}^{{\alpha }_0\left(x\right)-2}t+\rho \left(x\right)\left|t\right|(4t^3-2tcost-4sint)$$

with its primitive function

$$F(x,t)=\frac{\eta \left(x\right)}{{\alpha }_0\left(x\right)}{\left|t\right|}^{{\alpha }_0\left(x\right)}+\rho \left(x\right)\left(\frac{4}{5}{\left|t\right|}^5-2t\left|t\right|sint\right),$$

where $\eta \in L^{\tau (·)}\left({\mathbb{R}}^N\right)\cap L^{\infty }\left({\mathbb{R}}^N\right)$ is a positive function and $\rho \in C({\mathbb{R}}^N,\mathbb{R})\cap L^{\infty }\left({\mathbb{R}}^N\right)$ with $0<{inf}_{x\in {\mathbb{R}}^N}\rho \left(x\right)\le {sup}_{x\in {\mathbb{R}}^N}\rho \left(x\right)<\infty$. Set $\tilde{\omega }:={inf}_{x\in {\mathbb{R}}^N}\rho \left(x\right)$ for all $x\in {\mathbb{R}}^N$, then we have

$$\begin{array}{cc}\hfill & f(x,t)t-\nu F(x,t)\hfill \\ \hfill & =\eta \left(x\right)\left(1-\frac{\nu }{{\alpha }_0\left(x\right)}\right){\left|t\right|}^{{\alpha }_0\left(x\right)}\hfill \\ \hfill & +\rho \left(x\right)\left({4\left|t\right|}^5-{2\left|t\right|}^{3}cost-4t\left|t\right|sint-\frac{4}{5}{\nu \left|t\right|}^5+2\nu t\left|t\right|sint\right)\hfill \\ \hfill & \ge {\left|\right|\eta \left|\right|}_{L^{\infty }\left({\mathbb{R}}^N\right)}\left(1-\nu \right){\left|t\right|}^2+\tilde{\omega }\left({4\left|t\right|}^5-\frac{4}{5}{\nu \left|t\right|}^5-{2\left|t\right|}^{3}cost+(2\nu -4)t\left|t\right|sint\right)\hfill \\ \hfill & \ge {min\left\{\right|\left|\eta \right||}_{L^{\infty }\left({\mathbb{R}}^N\right)},\tilde{\omega }\}\left({4\left|t\right|}^3-\frac{4}{5}{\nu \left|t\right|}^3-2\left|t\right|cost-(3\nu -3)\right)t^2\hfill \\ \hfill & \ge {min\left\{\right|\left|\eta \right||}_{L^{\infty }\left({\mathbb{R}}^N\right)},\tilde{\omega }\}\left({\left|t\right|}^3+\left(3-\frac{4}{5}\nu \right){\left|t\right|}^3-2\left|t\right|-(3\nu -3)\right)\hfill \\ \hfill & \ge {min\left\{\right|\left|\eta \right||}_{L^{\infty }\left({\mathbb{R}}^N\right)},\tilde{\omega }\}\left({\left|t\right|}^2-(3\nu -3)\right)\hfill \\ \hfill & \ge 0\hfill \end{array}$$

for $\left|t\right|\ge t_0$, where $t_0\ge 3$ is chosen such that $\left(3-\frac{4}{5}\nu \right)t_0^3-2t_0\ge 0$ and ν is a value that lies in the interval $(3,\frac{15}{4}]$. Hence,(H1)–(H4)are fulfilled.

Example 2.

If ${\alpha }_0\left(x\right)<\alpha \left(x\right)<s\left(x\right)$ for all $x\in {\mathbb{R}}^N$ and we consider the function

$$f(x,t)={\eta \left(x\right)\left|t\right|}^{{\alpha }_0\left(x\right)-2}t+\rho \left(x\right){\left|t\right|}^{s\left(x\right)-1}t\left[(s\left(x\right)+3)t^2-2(s\left(x\right)+2)\left|t\right|+(s\left(x\right)+1)\right]$$

with its primitive function

$$F(x,t)=\frac{\eta \left(x\right)}{{\alpha }_0\left(x\right)}{\left|t\right|}^{{\alpha }_0\left(x\right)}+{\rho \left(x\right)\left(\right|t|}^{s\left(x\right)+3}-{2\left|t\right|}^{s\left(x\right)+2}+{\left|t\right|}^{s\left(x\right)+1}),$$

where η and ρ are given in Example 1. Then, we have

$$\begin{array}{cc}\hfill & f(x,t)t-\nu F(x,t)\hfill \\ \hfill & \ge {\left|\right|\eta \left|\right|}_{L^{\infty }\left({\mathbb{R}}^N\right)}\left(1-\nu \right){\left|t\right|}^{s\left(x\right)+1}\hfill \\ \hfill & +\rho \left(x\right)\left({(s\left(x\right)+3-\nu )\left|t\right|}^{s\left(x\right)+3}-{2(s\left(x\right)+2-\nu )\left|t\right|}^{s\left(x\right)+2}+(s\left(x\right)+1-\nu ){\left|t\right|}^{s\left(x\right)+1}\right)\hfill \\ \hfill & \ge {\left|\right|\eta \left|\right|}_{L^{\infty }\left({\mathbb{R}}^N\right)}\left(1-\nu \right){\left|t\right|}^{s\left(x\right)+1}\hfill \\ \hfill & +\tilde{\omega }\left({(s\left(x\right)+3-\nu )\left|t\right|}^2-2(s\left(x\right)+2-\nu )\left|t\right|-(\nu -s_{-}-1)\right){\left|t\right|}^{s\left(x\right)+1}\hfill \\ \hfill & \ge {min\left\{\right|\left|\eta \right||}_{L^{\infty }\left({\mathbb{R}}^N\right)},\tilde{\omega }\}\left({\left|t\right|}^2+{(s\left(x\right)+2-\nu )\left(\right|t|}^2-2\left|t\right|)-(2\nu -s_{-}-2)\right){\left|t\right|}^{s\left(x\right)+1}\hfill \\ \hfill & \ge 0\hfill \end{array}$$

for $\left|t\right|\ge t_0$, where $t_0>1$ is chosen such that $t_0^2-2t_0\ge 0$, $\tilde{\omega }$ is given in Example 1 and ν is a value that lies in the interval $(s_{+},s\left(x\right)+2]$. Hence, this example satisfies the conditions(H1)–(H4).

Definition 2.

We say that $u\in X$ is a weak solution of the problem Equation (3) if

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\varphi (x,\nabla w)·\nabla vdx+& {\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\right|}^{\alpha \left(x\right)-2}wvdx\hfill \\ \hfill & =\lambda {\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w\right|}^{r\left(x\right)-2}wvdx+{\int }_{{\mathbb{R}}^N}f(x,w)vdx\hfill \end{array}$$

for any $v\in X$.

Let us define the functional $\mathcal{A}:X\to \mathbb{R}$ by

$$\mathcal{A}\left(w\right)={\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w)dx+{\int }_{{\mathbb{R}}^N}\frac{V\left(x\right)}{\alpha \left(x\right)}{\left|w\right|}^{\alpha \left(x\right)}dx.$$

Then, under assumptions (HA1)–(HA6) and (V), $\mathcal{A}$ is Fréchet differentiable on X, and its derivative is given by

$$⟨{\mathcal{A}}^{\prime }\left(w\right),v⟩={\int }_{{\mathbb{R}}^N}\varphi (x,\nabla w)·\nabla vdx+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\right|}^{\alpha \left(x\right)-2}wvdx,$$

see [5]. Under conditions (B1), (B2), (H1) and (H2), let us define the functional ${\mathcal{B}}_{\lambda }:X\to \mathbb{R}$ by

$${\mathcal{B}}_{\lambda }\left(w\right)=\lambda {\int }_{{\mathbb{R}}^N}\frac{\kappa \left(x\right)}{\gamma \left(x\right)}{\left|w\right|}^{\gamma \left(x\right)}dx+{\int }_{{\mathbb{R}}^N}F(x,w)dx$$

for any $\lambda >0$. Then, it follows from [5] that ${\mathcal{B}}_{\lambda }\in C^1(X,\mathbb{R})$ and its Fréchet derivative is

$$⟨{\mathcal{B}}_{\lambda }^{\prime }\left(w\right),v⟩=\lambda {\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w\right|}^{\gamma \left(x\right)-2}wvdx+{\int }_{{\mathbb{R}}^N}f(x,w)vdx$$

for any $w,v\in X$. Next, we define the functional ${\mathcal{J}}_{\lambda }:X\to \mathbb{R}$ by

$${\mathcal{J}}_{\lambda }\left(w\right)=\mathcal{A}\left(w\right)-{\mathcal{B}}_{\lambda }\left(w\right).$$

Then, it follows that the functional ${\mathcal{J}}_{\lambda }\in C^1(X,\mathbb{R})$ and its Fréchet derivative is

$$\begin{array}{cc}\hfill ⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w\right),v⟩& ={\int }_{{\mathbb{R}}^N}\varphi (x,\nabla w)·\nabla vdx+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\right|}^{\alpha \left(x\right)-2}wvdx\hfill \\ \hfill & -\lambda {\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w\right|}^{\gamma \left(x\right)-2}wvdx-{\int }_{{\mathbb{R}}^N}f(x,w)vdx\hfill \end{array}$$

for any $w,v\in X$.

Lemma 6

([5]). Assume the (HA1) – (HA6) and (V) are satisfied. Then, the functional $\mathcal{A}$ is convex, of class $C^1$, and sequentially weakly lower semicontinuous in X.

Lemma 7

([5]). Assume that (HA1)–(HA6) and (V) hold. Then, the operator ${\mathcal{A}}^{\prime }:X\to X^{*}$ has the following properties:

(1)

${\mathcal{A}}^{\prime }$ is a continuous, bounded and strictly monotone operator.

If(HA7)is also satisfied, we have

(2)

${\mathcal{A}}^{\prime }$ is a mapping of type $\left(S_{+}\right)$, that is, if $w_n\rightharpoonup w$ in X and

$$\underset{n\to \infty }{lim\; sup}⟨{\mathcal{A}}^{\prime }\left(w_n\right)-{\mathcal{A}}^{\prime }\left(w\right),w_n-w⟩\le 0,$$

then $w_n\to w$ in $X;$

(3)

${\mathcal{A}}^{\prime }$ is a homeomorphism.

We introduce the Cerami condition which is the compactness condition of the Palais–Smale type.

Definition 3.

Let E be a real Banach space with dual space $E^{*}$, let $\mathcal{J}:E\to \mathbb{R}$ be a functional of class $C^1(E,\mathbb{R})$. Then $\mathcal{J}$ fulfills the Cerami condition ($\left(C\right)$-condition for short) in E, if any $\left(C\right)$-sequence $\left\{u_n\right\}\subset E$, i.e., $\left\{\mathcal{J}\left(u_n\right)\right\}$ is bounded and $\left|\right|{\mathcal{J}}^{\prime }\left(u_n\right){\left|\right|}_{E^{*}}(1+||u_n{\left|\right|}_E)\to 0\mathrm{as}n\to \infty$, has a convergent subsequence in E. Also $\mathcal{J}$ satisfies the Cerami condition at level c (${\left(C\right)}_c$-condition, for short) in E if any ${\left(C\right)}_c$-sequence $\left\{u_n\right\}\subset E$, i.e., $\mathcal{J}\left(u_n\right)\to c$ as $n\to \infty$ and $\left|\right|{\mathcal{J}}^{\prime }\left(u_n\right){\left|\right|}_{E^{*}}(1+||u_n{\left|\right|}_E)\to 0$ as $n\to \infty$, has a convergent subsequence in E.

The basic idea of proof of this consequence follows the analogous arguments as in [38]; see also [31].

Lemma 8.

Assume that (HA1)–(HA7) , (V), (B1)–(B2), (H1)–(H2), (H4) hold. Then, for any $\lambda >0$, the functional ${\mathcal{J}}_{\lambda }$ satisfies the $\left(C\right)$-condition.

Proof. 

Let $\left\{w_n\right\}$ be a $\left(C\right)$-sequence in X, i.e., ${sup}_{n\in \mathbb{N}}\left|{\mathcal{J}}_{\lambda }\left(w_n\right)\right|\le {\mathcal{K}}_1$ and $⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right),w_n⟩=o\left(1\right)\to 0$, as $n\to \infty$, where ${\mathcal{K}}_1$ is a positive constant. First, we show that $\left\{w_n\right\}$ is a bounded sequence in X. To this end, suppose to the contrary that $\left|\right|w_n{\left|\right|}_X>1$ and $\left|\right|w_n{\left|\right|}_X\to \infty$ as $n\to \infty$. Denote $\{a\le |w_n|\le b\}:=\{x\in {\mathbb{R}}^N:a\le |w_n\left(x\right)|\le b\}$ for any real number a and b. Since $V\left(x\right)\to +\infty$ as $\left|x\right|\to \infty$, then

$$\begin{array}{cc}\hfill \left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx& -C_4{\int }_{|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+{\sigma }_1\left(x\right)|w_n|+{\sigma }_2{\left|w_n\right|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx-{\mathcal{K}}_0\hfill \end{array}$$

(5)

for any positive constant $C_4$ and for some positive constant ${\mathcal{K}}_0$. In fact, by Young’s inequality we know that

$$\begin{array}{cc}\hfill & \left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx-C_4{\int }_{|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+{\sigma }_1\left(x\right)|w_n|+{\sigma }_2{\left|w_n\right|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \ge \left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4{\int }_{|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+{\sigma }_1^{{\ell }^{\prime }\left(x\right)}\left(x\right)+|w_n{|}^{\ell \left(x\right)}+{\sigma }_2{\left|w_n\right|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx+\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{|w_n|\le M}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4{\int }_{|w_n|\le 1}|w_n{|}^{\alpha \left(x\right)}+|w_n{|}^{\ell \left(x\right)}+{\sigma }_2{|w_n|}^{\ell \left(x\right)}dx\hfill \\ \hfill & -C_4{\int }_{1<|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+|w_n{|}^{\ell \left(x\right)}+{\sigma }_2|w_n{|}^{\ell \left(x\right)}dx-C_4\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx+\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{|w_n|\le M}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4(2+{\sigma }_2){\int }_{|w_n|\le 1}{\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4(1+M^{{\ell }_{+}-{\alpha }_{-}}({\sigma }_2+1)){\int }_{1<|w_n|\le M}{\left|w_n\right|}^{\alpha \left(x\right)}dx-{\tilde{C}}_4\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx+\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{|w_n|\le M}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4{M}^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2){\int }_{|w_n|\le M}{\left|w_n\right|}^{\alpha \left(x\right)}dx-{\tilde{C}}_4,\hfill \end{array}$$

where $C_4$ and ${\tilde{C}}_4$ are positive constants. Because $V\left(x\right)\to +\infty$ as $\left|x\right|\to \infty$, there exists $r_0>r$ such that $V\left(x\right)\ge 2C_4\left(M^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2)\right)\frac{\nu {\alpha }_{+}}{\nu -{\alpha }_{+}}$ for every $x\in {\mathbb{R}}^N$ with $\left|x\right|\ge r_0$. Then, we know that

$$V\left(x\right)|w_n{|}^{\alpha \left(x\right)}\ge 2C_4\left(M^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2)\right)\frac{\nu {\alpha }_{+}}{\nu -{\alpha }_{+}}{\left|w_n\right|}^{\alpha \left(x\right)}$$

for $\left|x\right|\ge r_0$. Set $B_r:=\{x\in {\mathbb{R}}^N:\left|x\right|\le r\}$, ${\mathsf{\Gamma }}_0=\left\{\right|w_n|\le M\}\cap B_{r_0}$ and ${\mathsf{\Gamma }}_1=\left\{\right|w_n|\le M\}\cap B_{r_0}^c$. Since $V\left(x\right)\in L_{loc}^1\left({\mathbb{R}}^N\right)$, we infer

$${\int }_{{\mathsf{\Gamma }}_0}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx\le C_5\mathrm{and}{\int }_{{\mathsf{\Gamma }}_0}{\left|w_n\right|}^{\alpha \left(x\right)}dx\le C_6$$

for some positive constants $C_5$ and $C_6$. This yields

$$\begin{array}{cc}\hfill & \left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx-C_4{\int }_{|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+{\sigma }_1\left(x\right)|w_n|+{\sigma }_2{|w_n|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx+\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathsf{\Gamma }}_1}V\left(x\right){|w_n|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & +\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathsf{\Gamma }}_0}V\left(x\right)|w_n{|}^{\alpha \left(x\right)}dx-C_4\left(M^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2)\right){\int }_{{\mathsf{\Gamma }}_1}{|w_n|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_4\left(M^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2)\right){\int }_{{\mathsf{\Gamma }}_0}{|w_n|}^{\alpha \left(x\right)}dx-{\tilde{C}}_4\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){|w_n|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & +{\int }_{{\mathsf{\Gamma }}_1}\left[\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)V\left(x\right)-C_4\left(M^{{\ell }_{+}-{\alpha }_{-}}(2+{\sigma }_2)\right)\right]{|w_n|}^{\alpha \left(x\right)}dx-{\mathcal{K}}_0\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){|w_n|}^{\alpha \left(x\right)}dx-{\mathcal{K}}_0,\hfill \end{array}$$

as required. From (HA6) and (H2), one has

$$\begin{array}{cc}\hfill & {\mathcal{K}}_1+o\left(1\right)\hfill \\ \hfill & \ge {\mathcal{J}}_{\lambda }\left(w_n\right)-\frac{1}{\nu }⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right),w_n⟩\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w_n)dx+{\int }_{{\mathbb{R}}^N}\frac{V\left(x\right)}{\alpha \left(x\right)}|w_n{|}^{\alpha \left(x\right)}dx-\lambda {\int }_{{\mathbb{R}}^N}\frac{\kappa \left(x\right)}{\gamma \left(x\right)}{|w_n|}^{\gamma \left(x\right)}dx\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}F(x,w_n)dx-\frac{1}{\nu }{\int }_{{\mathbb{R}}^N}\varphi (x,\nabla w_n)·\nabla w_{n}dx-\frac{1}{\nu }{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & +\frac{\lambda }{\nu }{\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx+\frac{1}{\nu }{\int }_{{\mathbb{R}}^N}f(x,w_n)w_{n}dx\hfill \\ \hfill & \ge {\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w_n)dx-\frac{1}{\nu }{\int }_{{\mathbb{R}}^N}\varphi (x,\nabla w_n)·\nabla w_{n}dx+\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx+{\int }_{{\mathbb{R}}^N}\frac{1}{\nu }f(x,w_n)w_n-F(x,w_n)dx\hfill \\ \hfill & \ge {\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w_n)dx-\frac{1}{\nu }{\int }_{{\mathbb{R}}^N}p\left(x\right)\Phi (x,\nabla w_n)dx+\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx+{\int }_{|w_n|>M}\frac{1}{\nu }f(x,w_n)w_n-F(x,w_n)dx\hfill \\ \hfill & -C_4{\int }_{|w_n|\le M}|w_n{|}^{\alpha \left(x\right)}+{\sigma }_1\left(x\right)|w_n|+{\sigma }_2{\left|w_n\right|}^{\ell \left(x\right)}dx.\hfill \end{array}$$

Combining this with Equation (5), (HA5), (H4) and Proposition 1, we derive that

$$\begin{array}{cc}\hfill & {\mathcal{K}}_1+o\left(1\right)\hfill \\ \hfill & \ge min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\left({\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w_n)dx+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\right)\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx-{\mathcal{K}}_0\hfill \\ \hfill & \ge min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\hfill \\ \hfill & \times \left({\int }_{{\mathsf{\Lambda }}_{\nabla w_n}}\Phi (x,\nabla w_n)dx+{\int }_{{\mathsf{\Lambda }}_{\nabla w_n}^c}\Phi (x,\nabla w_n)dx+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\right)\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx-{\mathcal{K}}_0\hfill \\ \hfill & \ge min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\hfill \\ \hfill & \times \left(C_1\left({\int }_{{\mathsf{\Lambda }}_{\nabla w_n}}|\nabla w_n{|}^{r\left(x\right)}dx+{\int }_{{\mathsf{\Lambda }}_{\nabla w_n}^c}{|\nabla w_n|}^{s\left(x\right)}dx\right)+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w_n\right|}^{\alpha \left(x\right)}dx\right)\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx-{\mathcal{K}}_0\hfill \\ \hfill & \ge min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\hfill \\ \hfill & \times (\frac{C_1}{2^{s_{+}}}min\left\{\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{-}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{+}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{s_{-}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{s_{+}}\right\}\hfill \\ \hfill & +min\left\{\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{-}},\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{+}}\right\})\hfill \\ \hfill & -\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\int }_{{\mathbb{R}}^N}\kappa \left(x\right){\left|w_n\right|}^{\gamma \left(x\right)}dx-{\mathcal{K}}_0,\hfill \end{array}$$

where p is Lipschitz continuous which satisfies $s(·)\le p(·)\ll \nu$ and $\nu$ is a positive constant from (H4). Since $\left|\right|w_n{\left|\right|}_X>1$ and $\left|\right|w_n{\left|\right|}_X\to \infty$ as $n\to \infty$, there are three cases to consider:

(i)

$\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}\ge 1$ and $\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}\ge 1$;

(ii)

$\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}<1$ and $\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}\ge 1$;

(iii)

$\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}\ge 1$ and $\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}<1$.

Denote

$$\begin{array}{cc}\hfill I_1:=min\{& \left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{-}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{+}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{s_{-}},\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{s_{+}}\}\hfill \end{array}$$

and

$$I_2:=min\left\{\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{-}},\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{+}}\right\}.$$

If (i) holds, then

$$\begin{array}{cc}\hfill I_1+I_2& =\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{-}}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{-}}\hfill \\ \hfill & \ge \left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{min\{{\alpha }_{-},r_{-}\}}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{min\{{\alpha }_{-},r_{-}\}}\hfill \\ \hfill & \ge C_7{\left(\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}\right)}^{min\{{\alpha }_{-},r_{-}\}}\hfill \\ \hfill & =C_7\left|\right|w_n{\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}.\hfill \end{array}$$

Suppose now (ii) is valid. We have

$$\begin{array}{cc}\hfill I_1+I_2& =\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{s_{+}}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{-}}\hfill \\ \hfill & \ge C_8{\left(\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}\right)}^{{\alpha }_{-}}\hfill \\ \hfill & =C_8\left|\right|w_n{\left|\right|}_X^{{\alpha }_{-}}.\hfill \end{array}$$

The case (iii) is similar to the case (ii). We have

$$\begin{array}{cc}\hfill I_1+I_2& =\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}^{r_{-}}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{+}}\hfill \\ \hfill & \ge C_9{\left(\left|\right|\nabla w_n{\left|\right|}_{L^{r(·),s(·)}\left({\mathbb{R}}^N\right)}+\left|\right|w_n{\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}\right)}^{min\{{\alpha }_{+},r_{-}\}}\hfill \\ \hfill & \ge C_9\left|\right|w_n{\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}.\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill {\mathcal{K}}_1+o\left(1\right)\ge & C_{10}min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\left|\right|w_n{\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}\hfill \\ \hfill & -2\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}max\left\{\right||w_n{\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{-}},\left|\right|w_n{\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{+}}\}-{\mathcal{K}}_0,\hfill \end{array}$$

for positive constant $C_{10}$. From this, we infer

$$\begin{array}{cc}\hfill {\mathcal{K}}_1+o\left(1\right)+{\mathcal{K}}_0& +2\lambda \left(\frac{1}{{\gamma }_{-}}-\frac{1}{\nu }\right){\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}max\left\{\right||w_n{\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{-}},\left|\right|w_n{\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{+}}\}\hfill \\ \hfill & \ge C_{10}min\left\{\left(1-\frac{p_{+}}{\nu }\right),\frac{1}{2}\left(\frac{1}{{\alpha }_{+}}-\frac{1}{\nu }\right)\right\}\left|\right|w_n{\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}.\hfill \end{array}$$

Since ${\gamma }_{+}<min\{{\alpha }_{-},r_{-}\}$, we arrive that $\left\{w_n\right\}$ is a bounded sequence in X and thus $\left\{w_n\right\}$ has a weakly convergent subsequence in X. Without loss of generality, we suppose that

$$w_n\rightharpoonup w_0\mathrm{in}X\mathrm{as}n\to \infty .$$

By Lemma 3.3 of [45], we infer that ${\mathcal{B}}_{\lambda }^{\prime }$ is compact, and so ${\mathcal{B}}_{\lambda }^{\prime }\left(w_n\right)\to {\mathcal{B}}_{\lambda }^{\prime }\left(w_0\right)$ in X as $n\to \infty$. Since ${\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right)\to 0$ as $n\to \infty$, we know that

$$⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right),w_n-w_0⟩\to 0\mathrm{and}⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_0\right),w_n-w_0⟩\to 0,$$

and thus

$$⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right)-{\mathcal{J}}_{\lambda }^{\prime }\left(w_0\right),w_n-w_0⟩\to 0$$

as $n\to \infty$. From this, we have

$$\begin{array}{cc}\hfill & ⟨{\mathcal{A}}^{\prime }\left(w_n\right)-{\mathcal{A}}^{\prime }\left(w_0\right),w_n-w_0⟩\hfill \\ \hfill & =⟨{\mathcal{B}}_{\lambda }^{\prime }\left(w_n\right)-{\mathcal{B}}_{\lambda }^{\prime }\left(w_0\right),w_n-w_0⟩+⟨{\mathcal{J}}_{\lambda }^{\prime }\left(w_n\right)-{\mathcal{J}}_{\lambda }^{\prime }\left(w_0\right),w_n-w_0⟩\to 0,\hfill \end{array}$$

namely, $⟨{\mathcal{A}}^{\prime }\left(w_n\right)-{\mathcal{A}}^{\prime }\left(w_0\right),w_n-w_0⟩\to 0$ as $n\to \infty$. Since X is reflexive by Proposition 2 and ${\mathcal{A}}^{\prime }$ is a mapping of type $\left(S_{+}\right)$ by Lemma 7 (2), we assert that

$$w_n\to w_0\mathrm{in}X\mathrm{as}n\to \infty .$$

The proof is completed □

Let $\mathfrak{W}$ be a reflexive and separable Banach space. Then it is known (see [52,53]) that there are $\left\{e_n\right\}\subseteq \mathfrak{W}$ and $\left\{h_n^{*}\right\}\subseteq {\mathfrak{W}}^{*}$ such that

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

and

$$\begin{array}{c}\hfill ⟨h_i^{*},e_j⟩=\left\{\begin{array}{cc}1\hfill & ifi=j\hfill \\ 0\hfill & ifi\ne j.\hfill \end{array}\right.\end{array}$$

Let us denote ${\mathfrak{W}}_n=\mathrm{span}\left\{e_n\right\}$, $Y_k={⨁}_{n=1}^k{\mathfrak{W}}_n$, and $Z_k=\overline{{⨁}_{n=k}^{\infty }{\mathfrak{W}}_n}$.

Definition 4.

Let E be a real separable and reflexive Banach space. We say that $\mathcal{J}$ satisfies the ${\left(C\right)}_c^{*}$-condition (with respect to $Y_n$) if any sequence $\left\{w_n\right\}\subset E$ for which $w_n\in Y_n$, for any $n\in \mathbb{N}$,

$$\mathcal{J}\left(w_n\right)\to c\mathrm{and}\left|\right|{\left(\mathcal{J}{|}_{Y_n}\right)}^{\prime }\left(w_n\right){\left|\right|}_{E^{*}}(1+||w_n{\left|\right|}_E)\to 0\mathrm{as}n\to \infty ,$$

contains a subsequence converging to a critical point of $\mathcal{J}$.

Lemma 9.

Let E be a reflexive Banach space and let $\mathcal{J}\in C^1(E,\mathbb{R})$ be an even functional. If there exists $k_0>0$ such that, for each $k\ge k_0$, there are ${\varkappa }_k>d_k>0$ such that

(A1)

$inf\{\mathcal{J}\left(\upsilon \right):\upsilon \in Z_k,|\left|\upsilon \right|{|}_E={\varkappa }_k\}\ge 0$;

(A2)

${\beta }_k:=max\{\mathcal{J}\left(\upsilon \right):\upsilon \in Y_k{,\left|\right|\upsilon \left|\right|}_E=d_k\}<0$;

(A3)

${\zeta }_k:=inf\{\mathcal{J}\left(\upsilon \right):\upsilon \in Z_k{,\left|\right|\upsilon \left|\right|}_E\le {\varkappa }_k\}\to 0$ as $k\to \infty$;

(A4)

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

then $\mathcal{J}$ has a sequence of negative critical values converging to 0.

Lemma 10.

Let (HA1)–(HA7), (V), (B1)–(B2), (H1)–(H2), (H4) hold. Then ${\mathcal{J}}_{\lambda }$ satisfies the ${\left(C\right)}_c^{*}$-condition.

Proof. 

Since X is a reflexive Banach space in view of Proposition 2, the proof is quite close to that of Lemma 3.12 [37]. □

Now we are ready to prove our main result.

Theorem 1.

Assume that (HA1)–(HA6), (V), (B1)–(B2), (H1)–(H4) hold. If $f(x,-t)=-f(x,t)$ holds for all $(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$, then the problem Equation (3) has a sequence of nontrivial solutions $\left\{w_n\right\}$ in X such that ${\mathcal{J}}_{\lambda }\left(w_n\right)\to 0$ as $n\to \infty$ for any $\lambda >0$.

Proof. 

Since f is odd in $t\in \mathbb{R}$, it is obvious that ${\mathcal{J}}_{\lambda }$ is even. According to Lemma 10, ${\mathcal{J}}_{\lambda }$ ensures the ${\left(C\right)}_c^{*}$-condition for every $c\in \mathbb{R}$. Moreover, in view of Proposition 2 and Remark 2.7 in [6], we infer that X is a separable and reflexive Banach space. Now, let us check that conditions (A1), (A2) and (A3) of Lemma 9 are satisfied.

(A1): For convenience, we denote

$${\chi }_{1,k}=\underset{{\left|\right|w\left|\right|}_X=1,w\in Z_k}{sup}{\left|\right|w\left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)},{\chi }_{2,k}=\underset{{\left|\right|w\left|\right|}_X=1,w\in Z_k}{sup}{\left|\right|w\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}$$

and

$${\chi }_k=max\{{\chi }_{1,k},{\chi }_{2,k}\}.$$

Then, it can be easily validate that ${\chi }_{1,k}\to 0$ and ${\chi }_{2,k}\to 0$ as $k\to \infty$ (see [54]) and therefore ${\chi }_k\to 0$ as $k\to \infty$. Hence, we derive that ${\chi }_k<1$ for k large enough. Moreover, it is derived in the same way as before,

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\lambda }\left(w\right)& ={\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w)dx+{\int }_{{\mathbb{R}}^N}\frac{V\left(x\right)}{\alpha \left(x\right)}{\left|w\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -\lambda {\int }_{{\mathbb{R}}^N}\frac{\kappa \left(x\right)}{\gamma \left(x\right)}{\left|w\right|}^{\gamma \left(x\right)}dx-{\int }_{{\mathbb{R}}^N}F(x,w)dx\hfill \\ \hfill & \ge C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\left|\right|w\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}\hfill \\ \hfill & -\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}max\left\{{\left|\right|w\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{-}}{,\left|\right|w\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{+}}\right\}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\left|\right|w\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}-\frac{{\sigma }_2}{{\ell }^{-}}{\left|\right|w\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}^{\ell +}\hfill \\ \hfill & \ge C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\left|\right|w\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}-\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}{\chi }_k^{{\gamma }_{-}}{\left|\right|w\left|\right|}_X^{{\gamma }_{+}}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\chi }_k{\left|\right|w\left|\right|}_X-\frac{{\sigma }_2}{{\ell }^{-}}{\chi }_k{\left|\right|w\left|\right|}_X^{\ell +}\hfill \\ \hfill & \ge C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\left|\right|w\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}-\left(\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{{\sigma }_2}{{\ell }_{-}}\right){\chi }_k^{{\gamma }_{-}}{\left|\right|w\left|\right|}_X^{2{\ell }_{+}}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\chi }_k{\left|\right|w\left|\right|}_X.\hfill \end{array}$$

for ${\left|\right|w\left|\right|}_X\ge 1$, where $C_{11}$ is positive constant. Set

$${\varkappa }_k={\left[\left(\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}\right)\frac{{\chi }_k^{{\gamma }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{1}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}.$$

(6)

Let $w\in Z_k$ with ${\left|\right|w\left|\right|}_X={\varkappa }_k>1$ for sufficiently large k. Then, because

$$\underset{k\to \infty }{lim}\frac{C_{11}}{2}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\varkappa }_k^{min\{{\alpha }_{-},r_{-}\}}=\infty ,$$

there exists $K\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\lambda }\left(w\right)& \ge C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\left|\right|w\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}-\left(\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{{\sigma }_2}{{\ell }_{-}}\right){\chi }_k^{{\gamma }_{-}}{\left|\right|w\left|\right|}_X^{2{\ell }_{+}}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\chi }_k{\left|\right|w\left|\right|}_X\hfill \\ \hfill & \ge \frac{C_{11}}{2}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}{\varkappa }_k^{min\{{\alpha }_{-},r_{-}\}}-2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}}{\chi }_k{\varkappa }_k\ge 0\hfill \end{array}$$

for all $k\in \mathbb{N}$ with $k\ge K$. Consequently,

$$inf\{{\mathcal{J}}_{\lambda }\left(w\right):w\in Z_k,|\left|w\right|{|}_X={\varkappa }_k\}\ge 0.$$

(A2): Since $Y_k$ is finite dimensional, it is clear that all the norms are equivalent. Then, there exist positive constants ${\vartheta }_{1,k}$, ${\vartheta }_{2,k}$ and ${\vartheta }_{3,k}$ such that

$${\vartheta }_{1,k}{\left|\right|w\left|\right|}_X\le {\left|\right|w\left|\right|}_{L^{{\alpha }_0(·)}(\eta ,{\mathbb{R}}^N)}{\mathrm{and}\left|\right|w\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}\le {\vartheta }_{2,k}{\left|\right|w\left|\right|}_X$$

and

$${\left|\right|\nabla w\left|\right|}_{L^{r(·)}\left({\mathsf{\Lambda }}_{\nabla w}\right)}+{\left|\right|\nabla w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_{\nabla w}^c\right)}\le {\vartheta }_{3,k}{\left|\right|w\left|\right|}_X$$

for any $w\in Y_k$. From (H2) and (H3), there exist positive constants $C_{12}$ and $C_{13}$ such that

$$F(x,t)\ge C_{12}{\eta \left(x\right)\left|t\right|}^{{\alpha }_0\left(x\right)}-C_{13}{\left|t\right|}^{\ell \left(x\right)}$$

for almost all $(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$. Let $w\in Y_k$ with ${\left|\right|w\left|\right|}_X\le 1$. Then, we have

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\lambda }\left(w\right)& ={\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w)dx+{\int }_{{\mathbb{R}}^N}\frac{V\left(x\right)}{\alpha \left(x\right)}{\left|w\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -\lambda {\int }_{{\mathbb{R}}^N}\frac{\kappa \left(x\right)}{\gamma \left(x\right)}{\left|w\right|}^{\gamma \left(x\right)}dx-{\int }_{{\mathbb{R}}^N}F(x,w)dx\hfill \\ \hfill & \le {\int }_{{\mathsf{\Lambda }}_{\nabla w}}\Phi (x,\nabla w)dx+{\int }_{{\mathsf{\Lambda }}_{\nabla w}^c}\Phi (x,\nabla w)dx+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_{12}{\int }_{{\mathbb{R}}^N}{\eta \left(x\right)\left|w\right|}^{{\alpha }_0\left(x\right)}dx+C_{13}{\int }_{{\mathbb{R}}^N}{\left|w\right|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \le C_2\left({\int }_{{\mathsf{\Lambda }}_{\nabla w}}{|\nabla w|}^{r\left(x\right)}dx+{\int }_{{\mathsf{\Lambda }}_{\nabla w}^c}{|\nabla w|}^{s\left(x\right)}dx\right)+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|w\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -C_{12}{\int }_{{\mathbb{R}}^N}{\eta \left(x\right)\left|w\right|}^{{\alpha }_0\left(x\right)}dx+C_{13}{\int }_{{\mathbb{R}}^N}{\left|w\right|}^{\ell \left(x\right)}dx\hfill \\ \hfill & \le C_2\left({max\left\{\right||\nabla w||}_{L^{r(·)}\left({\mathsf{\Lambda }}_{\nabla w}\right)}^{r_{+}}{,\left|\right|\nabla w\left|\right|}_{L^{r(·)}\left({\mathsf{\Lambda }}_{\nabla w}\right)}^{r_{-}}{\}+max\{\left|\right|\nabla w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_{\nabla w}^c\right)}^{s_{+}}{,\left|\right|\nabla w\left|\right|}_{L^{s(·)}\left({\mathsf{\Lambda }}_{\nabla w}^c\right)}^{s_{-}}\}\right)\hfill \\ \hfill & +{\left|\right|w\left|\right|}_{L_V^{\alpha (·)}\left({\mathbb{R}}^N\right)}^{{\alpha }_{-}}-C_{12}{min\left\{\right|\left|w\right||}_{L^{{\alpha }_0(·)}(\eta ,{\mathbb{R}}^N)}^{{{\alpha }_0}_{+}}{,\left|\right|w\left|\right|}_{L^{{\alpha }_0(·)}(\eta ,{\mathbb{R}}^N)}^{{{\alpha }_0}_{-}}\}\hfill \\ \hfill & +C_{13}{max\left\{\right|\left|w\right||}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}^{{\ell }_{+}}{,\left|\right|w\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}^{{\ell }_{-}}\}\hfill \\ \hfill & \le 2C_{2}max\{{\vartheta }_{3,k}^{r_{-}},{\vartheta }_{3,k}^{s_{+}}\}{\left|\right|w\left|\right|}_X^{r_{-}}+{\left|\right|w\left|\right|}_X^{{\alpha }_{-}}-C_{12}min\{{\vartheta }_{1,k}^{{{\alpha }_0}_{-}},{\vartheta }_{1,k}^{{{\alpha }_0}_{+}}\}{\left|\right|w\left|\right|}_X^{{{\alpha }_0}_{+}}\hfill \\ \hfill & +C_{13}max\{{\vartheta }_{2,k}^{{\ell }_{-}},{\vartheta }_{2,k}^{{\ell }_{+}}\}{\left|\right|w\left|\right|}_X^{{\ell }_{-}}\hfill \\ \hfill & \le C_{14}{\left|\right|w\left|\right|}_X^{min\{{\alpha }_{-},r_{-}\}}-C_{12}min\{{\vartheta }_{1,k}^{{{\alpha }_0}_{-}},{\vartheta }_{1,k}^{{{\alpha }_0}_{+}}\}{\left|\right|w\left|\right|}_X^{{{\alpha }_0}_{+}}+C_{13}max\{{\vartheta }_{2,k}^{{\ell }_{-}},{\vartheta }_{2,k}^{{\ell }_{+}}\}{\left|\right|w\left|\right|}_X^{{\ell }_{-}}\hfill \end{array}$$

for a positive constant $C_{14}$. Let $\hslash \left(t\right)=C_{14}{t}^{min\{{\alpha }_{-},r_{-}\}}-C_{12}min\{{\vartheta }_{1,k}^{{{\alpha }_0}_{-}},{\vartheta }_{1,k}^{{{\alpha }_0}_{+}}\}{t}^{{{\alpha }_0}_{+}}$$+C_{13}max\{{\vartheta }_{2,k}^{{\ell }_{-}},{\vartheta }_{2,k}^{{\ell }_{+}}\}{t}^{{\ell }_{-}}$. Since ${{\alpha }_0}_{+}<min\{{\alpha }_{-},r_{-}\}<{\ell }_{-}$, we can choose positive constant $t_1<1$ such that $\hslash \left(t\right)<0$ for every $t\in (0,t_1)$. Hence ${\mathcal{J}}_{\lambda }\left(w\right)<0$ for all $w\in Y_k$ with ${\left|\right|w\left|\right|}_X=t_1$. Taking $d_k=t_1$ for all $k\in \mathbb{N}$, one has

$${\beta }_k:=max\{{\mathcal{J}}_{\lambda }\left(w\right):w\in Y_k{,\left|\right|w\left|\right|}_X=d_k\}<0.$$

If necessary, we can make K larger, so that ${\varkappa }_k>d_k>0$ for every $k\ge K$.

(A3): Because $Y_k\cap Z_k\ne \varphi$ and $0<d_k<{\varkappa }_k$, we know ${\zeta }_k\le {\beta }_k<0$ for every $k\ge K$. For any $w\in Z_k$ with ${\left|\right|w\left|\right|}_X=1$ and $0<t<{\varkappa }_k$, we have

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\lambda }\left(tw\right)& ={\int }_{{\mathbb{R}}^N}\Phi (x,\nabla tw)dx+{\int }_{{\mathbb{R}}^N}\frac{V\left(x\right)}{\alpha \left(x\right)}{\left|tw\right|}^{\alpha \left(x\right)}dx\hfill \\ \hfill & -\lambda {\int }_{{\mathbb{R}}^N}\frac{\kappa \left(x\right)}{\gamma \left(x\right)}{\left|tw\right|}^{\gamma \left(x\right)}dx-{\int }_{{\mathbb{R}}^N}F(x,tw)dx\hfill \\ \hfill & \ge C_{1}max\left\{{\int }_{{\mathsf{\Lambda }}_{\nabla tw}^c}{|\nabla tw|}^{s\left(x\right)}dx,{\int }_{{\mathsf{\Lambda }}_{\nabla tw}}{|\nabla tw|}^{r\left(x\right)}dx\right\}\hfill \\ & -\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}max\left\{{\left|\right|tw\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{-}}{,\left|\right|tw\left|\right|}_{L^{{\ell }_0(·)}\left({\mathbb{R}}^N\right)}^{{\gamma }_{+}}\right\}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\left|\right|tw\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}-\frac{{\sigma }_2}{{\ell }_{-}}{max\left\{\right|\left|tw\right||}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}^{{\ell }_{-}}{,\left|\right|tw\left|\right|}_{L^{\ell (·)}\left({\mathbb{R}}^N\right)}^{{\ell }_{+}}\}\hfill \\ \hfill & \ge -\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}{\varkappa }_k^{{\gamma }_{+}}{\chi }_k^{{\gamma }_{-}}-2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\varkappa }_k{\chi }_k-\frac{{\sigma }_2}{{\ell }_{-}}{\varkappa }_k^{{\ell }_{+}}{\chi }_k^{{\ell }_{-}},\hfill \end{array}$$

for large enough k. Therefore by the definition of ${\varkappa }_k$, one has

$$\begin{array}{cc}\hfill 0>{\zeta }_k& \ge -\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}{\varkappa }_k^{{\gamma }_{+}}{\chi }_k^{{\gamma }_{-}}-2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\varkappa }_k{\chi }_k-\frac{{\sigma }_2}{{\ell }_{-}}{\varkappa }_k^{{\ell }_{+}}{\chi }_k^{{\ell }_{-}}\hfill \\ \hfill & =-\frac{2\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}{\left[\frac{\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{{\gamma }_{+}}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}{\chi }_k^{\frac{{\gamma }_{-}({\gamma }_{+}+min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+})}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}\hfill \\ \hfill & -2\left|\right|{\sigma }_1{\left|\right|}_{L^{{\ell }^{\prime }(·)}\left({\mathbb{R}}^N\right)}{\left[\frac{\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{1}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}{\chi }_k^{\frac{{\gamma }_{-}+min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}\hfill \\ \hfill & -\frac{{\sigma }_2}{{\ell }_{-}}{\left[\frac{\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{{\ell }_{+}}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}{\chi }_k^{\frac{{\gamma }_{-}{\ell }_{+}+(min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}){\ell }_{-}}{min\{{\alpha }_{-},r_{-}\}-2{\ell }_{+}}}.\hfill \end{array}$$

(7)

Because ${\ell }_{+}>min\{{\alpha }_{-},r_{-}\}$, ${\gamma }_{+}+min\{{\alpha }_{-},r_{-}\}<2{\ell }_{+}$, ${\gamma }_{-}{\ell }_{+}+{\ell }_{-}min\{{\alpha }_{-},r_{-}\}<2{\ell }_{-}{\ell }_{+}$ and ${\chi }_k\to 0$ as $k\to \infty$, we arrive that ${lim}_{k\to \infty }{\zeta }_k=0$.

As a result, all conditions of Lemma 9 are verified, and therefore, the problem Equation (3) has a sequence of nontrivial solutions $\left\{w_n\right\}$ in X such that ${\mathcal{J}}_{\lambda }\left(w_n\right)\to 0$ as $n\to \infty$ for all $\lambda >0$. □

4. Conclusions

In this paper, on a class of nonlinear term f which is different from those investigated in [39,40,42,43,44,45], we give the existence results of multiple solutions via making use of the dual fountain theorem as primary tool. When we check all assumptions in the dual fountain theorem, the conditions on the nonlinear term f near zero and at infinity are crucial, however we derive our main result without assuming them. As mentioned in the Introduction, the proof of our main result is different from that of the recent works [6,37,41,46,47]. From the viewpoint of [6,37], the assumptions (f1) and (f2) play a crucial role in obtaining Theorem 1. Under these two assumptions, the existence of two sequences $0<d_k<{\varkappa }_k$ sufficiently large is established in the papers [6,37]. Unfortunately, by applying the analogous argument as in [37], we cannot show the property (A3) in Theorem 1. More precisely, if we replace ${\varkappa }_k$ in Equation (6) with

$${\tilde{\varkappa }}_k={\left[\left(\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}\right)\frac{{\chi }_k^{{\gamma }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{1}{min\{{\alpha }_{-},r_{-}\}-{\ell }_{+}}}$$

and ${\gamma }_{+}+min\{{\alpha }_{-},r_{-}\}<{\ell }_{+}$, then in relation Equation (7),

$${\tilde{\varkappa }}_k^{{\gamma }_{+}}{\chi }_k^{{\gamma }_{-}}={\left[\frac{\frac{4\lambda }{{\gamma }_{-}}{\left|\right|\kappa \left|\right|}_{L^{{\tilde{\ell }}_0(·)}\left({\mathbb{R}}^N\right)}+\frac{2{\sigma }_2}{{\ell }_{-}}}{C_{11}min\left\{\frac{C_1}{2^{s_{+}}},\frac{1}{{\alpha }_{+}}\right\}}\right]}^{\frac{{\gamma }_{+}}{min\{{\alpha }_{-},r_{-}\}-{\ell }_{+}}}{\chi }_k^{\frac{{\gamma }_{-}({\gamma }_{+}+min\{{\alpha }_{-},r_{-}\}-{\ell }_{+})}{min\{{\alpha }_{-},r_{-}\}-{\ell }_{+}}}\to \infty \mathrm{as}k\to \infty$$

and thus we cannot obtain the property (A3) in ${\tilde{\varkappa }}_k$. However, the authors in [6,31] overcome this difficulty from new setting for ${\varkappa }_k$. In contrast, the existence of two sequences $0<d_k<{\varkappa }_k\to 0$ as $k\to \infty$ is obtained in [41,46,47] when (f1) is satisfied. On the other hand, we get Theorem 1 when (f1) and (f2) are not assumed. This is a novelty of the present paper. Moreover, we address to the readers several comments and perspectives:

I. We point out that with a similar analysis our main consequence continues to hold when $\mathrm{div}\left(\varphi \right(x,\nabla w\left)\right)$ in Equation (3) is changed into any Kirchhoff type $M\left({\int }_{{\mathbb{R}}^N}\Phi (x,\nabla w)dx\right)$$\mathrm{div}\left(\varphi \right(x,\nabla w\left)\right)$ where the continuous function $M:R_0^{+}\to R^{+}$ satisfies following conditions:

(1)

There exist positive real number $a_0$ such that $M\left(s\right)\ge a_0$ and M is nondecreasing for all $s>0$.

(2)

There exist $\theta >1$ such that $\beta >N\theta$ and $M\left(s\right)/s^{\theta -1}$ is nonincreasing for $s>0$.

II. Under the assumption (H1)–(H4), a new research direction in strong relationship with several related applications is the study of critical double-phase-type equations

$$-\mathrm{div}\left(\varphi (x,\nabla w)\right)+{V\left(x\right)\left|w\right|}^{\alpha \left(x\right)-2}w=\lambda \kappa \left(x\right){\left|w\right|}^{\gamma \left(x\right)-2}w+f(x,w)\mathrm{in}{\mathbb{R}}^N$$

where $s\left(x\right)<\gamma \left(x\right)$ for all $x\in {\mathbb{R}}^N$ and $\{x\in {\mathbb{R}}^N:\gamma \left(x\right)=r^{*}\left(x\right)\}\ne \varnothing$.