New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

Abstract

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: $M\left({\left[u\right]}_{s,p}^p+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx\right)[{(-\Delta )}_p^{s}u+$${V\left(x\right)\left|u\right|}^{p-2}u]={\lambda \left|u\right|}^{q-2}u\ln \left|u\right|,x\in {\mathbb{R}}^N,\left(\mathrm{P}\right)$. In these equations, $\lambda \in \mathbb{R}\setminus \left\{0\right\},$$p\in (1,+\infty )$, $s\in (0,1),sp<N,$ $p_s^{*}={\displaystyle \frac{Np}{N-sp}}$, $M\left(\tau \right)=a+b{\tau }^{\theta -1}$, $a,b\in {\mathbb{R}}^{+},$ $1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$, $V\left(x\right)\in C({\mathbb{R}}^N,\mathbb{R})$ is a potential function and ${(-\Delta )}_p^s$ is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of $\lambda$. More precisely, (i) Equation (P) has one ground state solution for $\lambda >0$ and $p\theta <q<p_s^{*}$, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for $\lambda <0$ and $p<q<p_s^{*}$, with positive and negative corresponding energy values, respectively.

1. Introduction

In this paper, we consider the following fractional p-Laplacian Kirchhoff equations

$$M\left({\left[u\right]}_{s,p}^p+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx\right)\left[{(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u\right]={\lambda \left|u\right|}^{q-2}u\ln \left|u\right|,x\in {\mathbb{R}}^N,$$

(1)

where $\lambda \in \mathbb{R}\setminus \left\{0\right\},$$p\in (1,+\infty )$, $s\in (0,1),sp<N,$ $p<q<p_s^{*}={\displaystyle \frac{Np}{N-sp}}$, and the fractional p-Laplacian operator ${(-\Delta )}_p^s$ is defined on smooth functions by

$${(-\Delta )}_p^{s}u\left(x\right)=2\underset{ϵ\to 0^{+}}{\lim }{\int }_{{\mathbb{R}}^N\setminus B_{ϵ}\left(x\right)}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}(u\left(x\right)-u\left(y\right))}{{|x-y|}^{N+ps}}}dy,x\in {\mathbb{R}}^N,$$

where $B_{ϵ}\left(x\right)=\left\{y\in {\mathbb{R}}^N:|x-y|<ϵ\right\}.$ Moreover, $V\left(x\right)$ and $M\left(\tau \right)$ satisfy the following:

• $\left(V_1\right)$ $V\left(x\right)\in C({\mathbb{R}}^N,\mathbb{R})$ satisfies meas$\{x\in {\mathbb{R}}^N:V\left(x\right)\le C\}<+\infty$ for all $C\in \mathbb{R}$;
• $\left(M_1\right)$ Let $M\left(\tau \right)=a+b{\tau }^{\theta -1}$ with $a,b\in {\mathbb{R}}^{+}$ and $1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$ for all $\tau \ge 0$.

In recent years, more and more researchers have focused their attention to the study of fractional problems with nonlocal terms by variational methods, as these problems have strong physical backgrounds and have been applied to many fields such as population dynamics, continuum mechanics and phase transition phenomena; we refer the reader to [1,2,3,4] and the references therein for further details. The existence of ground state solutions for fractional p-Laplacian Schrödinger problems has been obtained by the Mountain Pass Theorem and Nehari manifold; the reader is referred to [5,6] for details. By using the Symmetric Mountain Pass Theorem, the existence of infinitely many high energy solutions for the fractional p-Laplacian Schrödinger–Kirchhoff problems has been obtained [7]. By using the Nehari manifold and fibering map, the existence of multiple solutions for the fractional p-Laplacian Schrödinger problems with logarithmic nonlinearity has been proved [8]. In particular, Truong [9] has proved the existence of two nontrivial solutions of the following Schrödinger equation using fibrering maps and the Nehari manifold:

$${(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u=h(x,u),x\in {\mathbb{R}}^N,$$

(2)

where $h(x,u)={\mathcal{l}K\left(x\right)\left|u\right|}^{p-2}u\ln \left|u\right|$, $\mathcal{l}>0$, $K\left(x\right)$ and $V\left(x\right)$ satisfy some assumptions. Truong [10] has obtained two nontrivial solutions of (2) for

$$h(x,u)={\mathcal{l}K\left(x\right)\left|u\right|}^{p-2}u\ln \left|u\right|+{\mu Q\left(x\right)\left|u\right|}^{p-2}u,$$

where $\mathcal{l},\mu >0$.

Pucci, Xiang and Zhang [11] have studied the following Schrödinger–Kirchhoff problem by using Ekeland’s variational principle and the Mountain Pass Theorem:

$$M\left({\left[u\right]}_{s,p}^p\right){(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u=h(x,u),x\in {\mathbb{R}}^N,$$

(3)

where $M\left(\tau \right)$ and $h(x,u)$ satisfy some assumptions. The existence of multiple solutions of (3) has also been established [11]. Liang, Pu and Rdulescu [12] have proved the existence of a nontrivial solution of (3) in the case of high perturbations of the logarithmic nonlinearity for $h(x,u)={\kappa \left|u\right|}^{r-2}u\ln \left|u\right|+{\left|u\right|}^{p_s^{*}-2}u,$ where $\kappa >0$ is large and $p\theta <r<p_s^{*}$.

It is worth noting that a typical example of M is given by $M\left(\tau \right)=a+b{\tau }^{\theta -1}$ with $\theta >1,a>0$ and $b\ge 0$ for all $\tau \ge 0$. Assume that M is of this type. Then problem (1) is said to be non-degenerate if $a>0$ and $b\ge 0$; while it is called degenerate if $a=0$ and $b>0.$

Xiang, Hu and Yang [13] have studied the fractional Kirchhoff problem (3) in $\Omega$ by the Nehari manifold approach, where $h(x,u)={K\left(x\right)\left|u\right|}^{\theta p-2}u\ln \left|u\right|+{\mu \left|u\right|}^{r-2}u$ for $\mu >0$, ${\min }_{x\in \Omega }K\left(x\right)>0$ and $M\left(\tau \right)=a+b{\tau }^{\theta -1}(a,b\in {\mathbb{R}}^{+},\theta >1)$; in other words, the problem is non-degenerate.

Karim, Thanh and Bayrami-Aminlouee [14] have obtained the existence of infinitely many solutions for a new class of Schrödinger–Kirchhoff-type equations of the form

$$M\left({\left[u\right]}_{s,p}^p+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx\right)\left[{(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u\right]=\lambda h(x,u),x\in {\mathbb{R}}^N,$$

(4)

where $h(x,u)$ satisfies some assumptions and the Kirchhoff functions may be sign-changing and degenerate.

Inspired by the above literature, we consider the fractional Kirchhoff equations (1) in ${\mathbb{R}}^N$ under the non-degenerate case by using Ekeland’s variational principle and fibering map. Refs. [9,10,11,12,13] investigate the logarithmic nonlinearity ${\lambda \left|u\right|}^{q-2}u\ln \left|u\right|$ for $\lambda >0$. In contrast to previous studies, we also establish a new result for the case $\lambda <0$ in Section 4. To prove the existence of the solution for the case $\lambda <0$, we utilize the least energy solution of the following problem: ${(-\Delta )}_p^{s}u+{V\left(x\right)\left|u\right|}^{p-2}u={\left|u\right|}^{q-2}u,x\in {\mathbb{R}}^N$, where $0<s<1$, $ps<N$ and $1<p<q<p_s^{*}$. Based on the conclusions in [6,15], the energy value m to the least energy solution of the above equation is given in Section 4 below. Let $\delta >0$ be a suitable parameter and take the following positive constant:

$${\lambda }^{*}=\max \{1,e(q-p)\}\left(a{\delta }^{p-q}+b{\delta }^{p\theta -q}\right){\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{q-p}{p}}}.$$

(5)

The following discussion is divided into two cases: $\lambda >0$ and $\lambda <-{\lambda }^{*}<0$. Now, we summarize our main results using Theorems 1 and 2 below.

Theorem 1.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $p<p\theta <q<p_s^{*}$, then problem (1) has one ground state solution for $\lambda >0,$ with a positive corresponding energy value.

Theorem 2.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $p<q<p_s^{*}$, then problem (1) has at least two nontrivial solutions for $\lambda <-{\lambda }^{*}<0$, with positive and negative corresponding energy values, respectively.

Remark 1.

In this paper, the logarithmic nonlinearity plays a very important role. On the one hand, even if $\lambda >0$, the nonlinearity ${\lambda \left|u\right|}^{q-2}u\ln u$ is sign-changing, which is different from the nonlinearity ${\lambda \left|u\right|}^{q-2}u$. On the other hand, if we replace ${\lambda \left|u\right|}^{q-2}u\ln u$ by ${\lambda \left|u\right|}^{q-2}u$ in Equation (1), then the result of Theorem 1 still holds, but the result of Theorem 2 cannot be obtained because the results of the above two theorems indicate Equation (1) is superlinear for $\lambda >0$ or locally superlinear and locally sublinear for $\lambda <0$.

Remark 2.

To prove the existence of multiple solutions, a key step is to establish that the sets ${\mathcal{M}}^{+}$ and ${\mathcal{M}}^{-}$ in the following Lemma 9 are nonempty. However, the approach used in this paper does not suffice to prove Lemma 9 for the degenerate case where $a=0$ and $b>0$. Addressing this case may require different methods and is left for future work.

The rest of the paper is organized as follows. In Section 2, we give some notations and preliminary lemmas to prepare for the proof of our main results. Section 3 and Section 4 give the proof of Theorems 1 and 2, respectively.

2. Some Notations and Preliminary Lemmas

Some notations are given below.

$$W^{s,p}\left({\mathbb{R}}^N\right):=\left\{u\in L^p\left({\mathbb{R}}^N\right):u\mathrm{is}\mathrm{measurable}\mathrm{and}{\left[u\right]}_{s,p}<\infty \right\}$$

denotes the fractional Sobolev space equipped with the norm

$${\parallel u\parallel }_W={\left({\left[u\right]}_{s,p}^p+{\left|u\right|}_p^p\right)}^{{\displaystyle \frac{1}{p}}},$$

where the Gagliardo seminorm

$${\left[u\right]}_{s,p}^p={\int \int }_{{\mathbb{R}}^{2N}}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+ps}}}dxdy$$

and

$${\left|u\right|}_p^p={\int }_{{\mathbb{R}}^N}{\left|u\right|}^{p}dx.$$

In this paper, under the assumption of $\left(V_1\right)$, define

$$X=\left\{u\in W^{s,p}\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx<\infty \right\}$$

with the norm

$${\parallel u\parallel }_X={\left({\left[u\right]}_{s,p}^p+{\left|V^{{\displaystyle \frac{1}{p}}}u\right|}_p^p\right)}^{{\displaystyle \frac{1}{p}}},$$

and $X^{*}$ is the dual space of the Sobolev space $X.$ In [11], the weak solutions of (1) correspond to the critical point of the energy functional $I\in C^1(X,\mathbb{R})$ defined by

$$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{p}}\widehat{M}\left({\parallel u\parallel }_X^p\right)-\lambda F(x,u),\end{array}$$

where $\widehat{M}\left(t\right)={\int }_0^{t}M\left(\tau \right)d\tau$ and

$$F(x,u)={\displaystyle \frac{1}{q}}{\int }_{{\mathbb{R}}^N}\left({\left|u\right|}^q\ln \left|u\right|-{\displaystyle \frac{1}{q}}{\left|u\right|}^q\right)dx.$$

Define

$$\begin{array}{c}\hfill \mathcal{M}=\{u\in X\setminus \left\{0\right\}:\Phi \left(u\right)\triangleq \langle I^{\prime }\left(u\right),u\rangle =0\},\end{array}$$

(6)

where

$$\Phi \left(u\right)=M\left({\parallel u\parallel }_X^p\right){\parallel u\parallel }_X^p-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx.$$

If $u\in \mathcal{M}$ and $M\left(\tau \right)=a+b{\tau }^{\theta -1}(a,b\in {\mathbb{R}}^{+})$, then

$$\begin{array}{ccc}\hfill & & \langle {\Phi }^{\prime }\left(u\right),u\rangle \hfill \\ & & ={(p-q)a\parallel u\parallel }_X^p+{(p\theta -q)b\parallel u\parallel }_X^{p\theta }-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\hfill \\ & & ={(p\theta -p)b\parallel u\parallel }_X^{p\theta }-(q-p)\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\hfill \\ & & ={(p-p\theta )a\parallel u\parallel }_X^p-(q-p\theta )\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx.\hfill \end{array}$$

(7)

To study the existence of multiple solutions by fibering maps, we divide $\mathcal{M}$ into ${\mathcal{M}}^{+}$, ${\mathcal{M}}^{-}$ and ${\mathcal{M}}^0$, based on the critical points of the fibering maps, which represent local minima, local maxima, and points of inflection, respectively, that is

$$\begin{array}{cc}\hfill & {\mathcal{M}}^{+}=\{u\in \mathcal{M}:\langle {\Phi }^{\prime }\left(u\right),u\rangle >0\},\\ & {\mathcal{M}}^{-}=\{u\in \mathcal{M}:\langle {\Phi }^{\prime }\left(u\right),u\rangle <0\},\\ & {\mathcal{M}}^0=\{u\in \mathcal{M}:\langle {\Phi }^{\prime }\left(u\right),u\rangle =0\}.\end{array}$$

(8)

Lemma 1

([16,17]). Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. Then $X\hookrightarrow L^{\beta }\left({\mathbb{R}}^N\right)$ are continuous embeddings for $\beta \in [p,p_s^{*}]$, that is, for all $u\in X$, there exists $S_{\beta }>0$ such that

$$\begin{array}{c}\hfill {\left|u\right|}_{\beta }\le S_{\beta }{\parallel u\parallel }_X,\mathrm{for}\beta \in [p,p_s^{*}],\end{array}$$

(9)

and $X\hookrightarrow L^{\beta }\left({\mathbb{R}}^N\right)$ are compact embeddings for $\beta \in [p,p_s^{*}).$

Lemma 2.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $\tilde{u}\notin {\mathcal{M}}^0$ is a local minimizer for I on $\mathcal{M}$, then $I^{\prime }\left(\tilde{u}\right)=0$ in $X^{*}$.

Proof. 

According to [18], there exists a Lagrange multiplier $\mu \in \mathbb{R}$ such that $I^{\prime }\left(\tilde{u}\right)=\mu {\xi }^{\prime }\left(\tilde{u}\right),$ where

$$\xi \left(u\right)={a\parallel u\parallel }_X^p+{b\parallel u\parallel }_X^{p\theta }-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx.$$

Since $\tilde{u}$ is a local minimizer for I on $\mathcal{M}$, then $\langle I^{\prime }\left(\tilde{u}\right),\tilde{u}\rangle =0$ implies that

$$\begin{array}{c}\hfill \mu \langle {\xi }^{\prime }\left(\tilde{u}\right),\tilde{u}\rangle =0.\end{array}$$

(10)

However, $\tilde{u}\notin {\mathcal{M}}^0;$ then, by (8),

$$\langle {\xi }^{\prime }\left(\tilde{u}\right),\tilde{u}\rangle =\langle {\Phi }^{\prime }\left(\tilde{u}\right),\tilde{u}\rangle \ne 0,$$

which, together with (10), gives $\mu =0$. Hence, $I^{\prime }\left(\tilde{u}\right)=0$ in $X^{*}$. □

Lemma 3.

Let $p<q<\kappa <p_s^{*}$ and $t>0$, then

(i) 

For all $\epsilon >0,$ there exists $C_{\epsilon }>0$ such that $|t^{q-1}\ln t|\le \epsilon |t|+C_{\epsilon }{\left|t\right|}^{\kappa -1};$

(ii) 

$t^{q-p}\left(\ln t-{\displaystyle \frac{1}{q}}\right)\ge -{\displaystyle \frac{e^{-\frac{p}{q}}}{q-p}};$

(iii) 

$t^{q-p_s^{*}}\ln t\le {\displaystyle \frac{1}{e(p_s^{*}-q)}}.$

Proof. 

(i)

Based on the properties of logarithmic functions, we can obtain the conclusion.

(ii)

Let $\varphi \left(t\right)=t^{q-p}\left(\ln t-{\displaystyle \frac{1}{q}}\right)$ for $t>0$, through simple calculations, we obtain that $t_{\min }=e^{-{\displaystyle \frac{p}{q(q-p)}}}$ is the minimum point of function $\varphi$ on $(0,+\infty )$, and $\varphi \left(t_{\min }\right)=-{\displaystyle \frac{e^{-\frac{p}{q}}}{q-p}}$.

(iii)

Let $\overline{\varphi }\left(t\right)=t^{q-p_s^{*}}\ln t$ for $t>0$, we obtain that $t_{\max }=e^{-{\displaystyle \frac{1}{q-p_s^{*}}}}$ is the maximum point of function $\overline{\varphi }$ on $(0,+\infty )$, and $\overline{\varphi }\left(t_{\max }\right)={\displaystyle \frac{1}{e(p_s^{*}-q)}}$.

□

Lemma 4.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. Then,

(i) 

If $p<p\theta <q<p_s^{*}$ and $\lambda >0,$ then I is bounded below and coercive on $\mathcal{M};$

(ii) 

If $p<q<p_s^{*}$ and $\lambda <0,$ then I is bounded below and coercive on $X.$

Proof. 

(i) For $u\in \mathcal{M}$. If $\lambda >0$ and $p<p\theta <q<p_s^{*}$, then

$$\begin{array}{ccc}\hfill & & I\left(u\right)-{\displaystyle \frac{1}{q}}\langle I^{\prime }\left(u\right),u\rangle \hfill \\ & & ={\displaystyle \frac{q-p}{pq}}{a\parallel u\parallel }_X^p+{\displaystyle \frac{q-p\theta }{pq}}{b\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{1}{q^2}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\hfill \\ & & \ge {\displaystyle \frac{q-p\theta }{pq}}{b\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{1}{q^2}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\hfill \\ & & \ge {\displaystyle \frac{q-p\theta }{pq}}b{\parallel u\parallel }_X^{p\theta }\hfill \\ & & >0.\hfill \end{array}$$

It is easy to see that I is bounded below and

$$I\left(u\right)\to {\infty \mathrm{as}\parallel u\parallel }_X\to \infty .$$

Then, I is bounded below and coercive on $\mathcal{M}.$

(ii) For $u\in X$. If $\lambda <0$ and $p<q<p_s^{*}$, then, by Lemmas 1 and 3 (ii),

$$\begin{array}{ccc}\hfill & & I\left(u\right)\ge {\displaystyle \frac{b}{p\theta }}{\parallel u\parallel }_X^{p\theta }-{\displaystyle \frac{1}{q}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx+{\displaystyle \frac{1}{q^2}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\hfill \\ & & ={\displaystyle \frac{b}{p\theta }}{\parallel u\parallel }_X^{p\theta }-{\displaystyle \frac{1}{q}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\left(\ln \left|u\right|-{\displaystyle \frac{1}{q}}\right)dx\hfill \\ & & \ge {\displaystyle \frac{b}{p\theta }}{\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{\lambda e^{-\frac{p}{q}}}{q(q-p)}}{\int }_{{\mathbb{R}}^N}{\left|u\right|}^{p}dx\hfill \\ & & \ge {\displaystyle \frac{b}{p\theta }}{\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{\lambda e^{-\frac{p}{q}}{S}_p^p}{q(q-p)}}{\parallel u\parallel }_X^p\hfill \\ & & ={\parallel u\parallel }_X^p\left({\displaystyle \frac{b}{p\theta }}{\parallel u\parallel }_X^{p\theta -p}+{\displaystyle \frac{\lambda e^{-\frac{p}{q}}{S}_p^p}{q(q-p)}}\right).\hfill \end{array}$$

From $\theta >1$, it is easy to see that I is bounded below and

$$I\left(u\right)\to {\infty \mathrm{as}\parallel u\parallel }_X\to \infty .$$

Then, I is bounded below and coercive on $X.$ □

Lemma 5.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $p<q<p_s^{*}$, then any bounded Palais–Smale sequence of $I\left(u\right)$ on X has a convergent subsequence.

Proof. 

Let $\left\{u_n\right\}\subset X$ be a bounded Palais–Smale sequence for I. Passing to a subsequence if necessary, we determine that there exists $u\in X$ such that

$$\begin{array}{cc}\hfill & u_n\rightharpoonup u\mathrm{in}X,\hfill \\ \hfill & u_n\to u\mathrm{in}{L}^{\beta }\left({\mathbb{R}}^N\right)\mathrm{for}\beta \in [p,p_s^{*}),\hfill \\ \hfill & u_n\left(x\right)\to u\left(x\right)\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^N.\hfill \end{array}$$

(11)

By Lemma 3 (i), for all $\epsilon >0$, there exists $C_{\epsilon }>0$ such that

$$|t^{q-1}\ln t|\le \epsilon |t|+C_{\epsilon }{\left|t\right|}^{\kappa -1}.$$

According to (11) and Hölder’s inequality, there exists a constant $C>0$ such that

$$\begin{array}{cc}\hfill & \left|{\int }_{{\mathbb{R}}^N}\left(|u_n{|}^{q-1}\ln |u_n{|-|u|}^{q-1}\ln \left|u\right|\right)(u_n-u)dx\right|\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^N}\left(\epsilon |u_n|+C_{\epsilon }|u_n{|}^{\kappa -1}+\epsilon \left|u\right|+C_{\epsilon }{\left|u\right|}^{\kappa -1}\right)(u_n-u)dx\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\epsilon \left(\right|u_n|+|u\left|\right)(u_n-u)dx+{\int }_{{\mathbb{R}}^N}{C}_{\epsilon }\left(\right|u_n{|}^{\kappa -1}+{\left|u\right|}^{\kappa -1})(u_n-u)dx\hfill \\ \hfill & \le 4\epsilon \left(|u_n{|}_2^2+{\left|u\right|}_2^2\right)+C_{\epsilon }\left(|u_n{|}_{\kappa }^{\kappa -1}+{\left|u\right|}_{\kappa }^{\kappa -1}\right){|u_n-u|}_{\kappa }\hfill \\ \hfill & \le C(\epsilon +|u_n-u{|}_{\kappa }),\hfill \end{array}$$

where $p<q<\kappa <p_s^{*}.$ Since

$$|u_n{-u|}_{\kappa }\to 0\mathrm{in}{L}^{\kappa }\left({\mathbb{R}}^N\right)\mathrm{for}\kappa \in [p,p_s^{*}),$$

then

$$\left|{\int }_{{\mathbb{R}}^N}\left(|u_n{|}^{q-1}\ln |u_n{|-|u|}^{q-1}\ln \left|u\right|\right)(u_n-u)dx\right|\to 0.$$

Next, let $\phi \in X$ be fixed, and define a linear functional

$$B_{\phi }\left(\omega \right)=\underset{{\mathbb{R}}^{2N}}{\int \int }{\displaystyle \frac{{|\phi \left(x\right)-\phi \left(y\right)|}^{p-2}(\phi \left(x\right)-\phi \left(y\right))}{{|x-y|}^{N+ps}}}(\omega \left(x\right)-\omega \left(y\right))dxdy,\omega \in X.$$

Since

$$\langle I^{\prime }\left(u_n\right)-I^{\prime }\left(u\right),u_n-u\rangle \to 0,$$

we get

$$\begin{array}{cc}\hfill & M\left({\left[u_n\right]}_{s,p}^p\right)(B_{u_n}(u_n-u)-B_u(u_n-u))\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}V\left(x\right)\left(|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)(u_n-u)dx\hfill \\ \hfill & =\langle I^{\prime }\left(u_n\right)-I^{\prime }\left(u\right),u_n-u\rangle +\left(M\left({\left[u\right]}_{s,p}^p\right)-M\left({\left[u_n\right]}_{s,p}^p\right)\right)B_u(u_n-u)\hfill \\ \hfill & +\lambda {\int }_{{\mathbb{R}}^N}\left(|u_n{|}^q\ln |u_n{|-|u|}^q\ln \left|u\right|\right)(u_n-u)dx,\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill & M\left({\left[u_n\right]}_{s,p}^p\right)(B_{u_n}(u_n-u)-B_u(u_n-u))\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}V\left(x\right)\left(|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)(u_n-u)dx\to 0\hfill \end{array}$$

as $n\to \infty .$ By convexity, we obtain

$$\underset{n\to \infty }{\lim }[B_{u_n}(u_n-u)-B_u(u_n-u)]=0$$

and

$$\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^N}V\left(x\right)\left(|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)(u_n-u)dx=0.$$

We complete the proof via the argument of the following two cases:

Case 1: When $p\ge 2$, according to the following Simon inequality

$${|\xi -\eta |}^p\le C_p\left[\left(\right|\xi {|}^{p-2}\xi -\left|\eta {|}^{p-2}\eta \right)(\xi -\eta )\right],\xi ,\eta \in {\mathbb{R}}^N,$$

where $p\ge 2$ and $C_p>0$, we get

$$\begin{array}{c}\hfill {[u_n-u]}_{s,p}^p\le C_p\left(B_{u_n}(u_n-u)-B_u(u_n-u)\right),\end{array}$$

(12)

and

$$\begin{array}{c}\hfill |V^{{\displaystyle \frac{1}{p}}}(u_n-u){|}_p^p\le C_p{\int }_{{\mathbb{R}}^N}V\left(x\right)\left(|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)(u_n-u)dx.\end{array}$$

(13)

By (12) and (13), we have that

$${[u_n-u]}_{s,p}^p\to 0\mathrm{as}n\to \infty ,$$

and

$$|V^{{\displaystyle \frac{1}{p}}}(u_n-u){|}_p^p\to 0\mathrm{as}n\to \infty .$$

Thus,

$$\parallel u_n{-u\parallel }_X={\left({[u_n-u]}_{s,p}^p+|V^{{\displaystyle \frac{1}{p}}}(u_n-u){|}_p^p\right)}^{{\displaystyle \frac{1}{p}}}\to 0\mathrm{as}n\to \infty .$$

Case 2: When $1<p<2$, we have the following inequalities:

$${(\mu +\nu )}^{{\displaystyle \frac{2-p}{2}}}\le {\mu }^{{\displaystyle \frac{2-p}{2}}}+{\nu }^{{\displaystyle \frac{2-p}{2}}}\mathrm{for}\mu ,\nu \ge 0,$$

and

$${|\xi -\eta |}^p\le c_p{\left[\right(\left|\xi \right|}^{p-2}\xi -{\left|\eta \right|}^{p-2}{\eta )·(\xi -\eta )]}^{{\displaystyle \frac{p}{2}}}\times {\left(\right|\xi |}^p+{\left|\eta \right|}^p{)}^{{\displaystyle \frac{2-p}{2}}}\mathrm{for}\xi ,\eta \in {\mathbb{R}}^N,$$

where $c_p>0$. Then, we deduce that

$$\begin{array}{c}\hfill {[u_n-u]}_{s,p}^p\le C{[B_{u_n}(u_n-u)-B_u(u_n-u)]}^{{\displaystyle \frac{p}{2}}}\end{array}$$

(14)

and

$$\begin{array}{c}\hfill |V^{{\displaystyle \frac{1}{p}}}(u_n-u){|}_p^p\le C{\left({\int }_{{\mathbb{R}}^N}V\left(x\right)\left(|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)(u_n-u)dx\right)}^{{\displaystyle \frac{p}{2}}},\end{array}$$

(15)

where C is a positive constant. Combing (14) and (15), we get that

$$|V^{{\displaystyle \frac{1}{p}}}(u_n-u){|}_p^p+{[u_n-u]}_{s,p}^p\to 0\mathrm{as}n\to \infty .$$

Thus, $\parallel u_n{-u\parallel }_X={\left({[u_n-u]}_{s,p}^p+{|V^{{\displaystyle \frac{1}{p}}}(u_n-u)|}_p^p\right)}^{{\displaystyle \frac{1}{p}}}\to 0\mathrm{as}n\to \infty .$ □

3. $\mathit{\lambda }>\mathbf{0}$: The Existence of Ground State Solution

In this paper, let $\lambda >0$ and $p\theta <q<p_s^{*}$. We prove that problem (1) has one ground state solution. Firstly, it is given that the Nehari manifold ${\mathcal{M}}^{-}$ is nonempty by Lemma 6.

Lemma 6.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $p\theta <q<p_s^{*}$ and $u\in X\setminus \left\{0\right\}$, there exists $t_0>0$ such that

$$t_{0}u\in {\mathcal{M}}^{-}\mathrm{for}\lambda >0.$$

Proof. 

For $u\in X\setminus \left\{0\right\}$. Let

$$g_u\left(t\right)=t^{p-q}{a\parallel u\parallel }_X^p+t^{p\theta -q}{b\parallel u\parallel }_X^{p\theta }-\lambda \ln t{\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx,t>0,$$

then

$$\begin{array}{c}\hfill g_u^{\prime }\left(t\right)=(p-q)t^{p-q-1}{a\parallel u\parallel }_X^p+(p\theta -q)t^{p\theta -q-1}{b\parallel u\parallel }_X^{p\theta }-\lambda t^{-1}{\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx.\end{array}$$

(16)

By simple calculations, we have that $t^{q+1}{g}_u^{\prime }\left(t\right)=\langle {\Phi }^{\prime }\left(tu\right),tu\rangle ,$ where the definition of ${\Phi }^{\prime }\left(u\right)$ is given in (7), and

$$\begin{array}{c}\hfill tu\in \mathcal{M}\iff g_u\left(t\right)-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx=0.\end{array}$$

(17)

To prove $tu\in {\mathcal{M}}^{+}$ (or $tu\in {\mathcal{M}}^{-})$, we only need to prove

$$\begin{array}{cc}\hfill & g_u\left(t\right)-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q\ln \left|u\right|dx=0\mathrm{and}{g}_u^{\prime }\left(t\right)>0(\mathrm{or}{g}_u^{\prime }\left(t\right)<0).\end{array}$$

(18)

If $\lambda >0$ and $p<p\theta <q<p_s^{*}$, then (16) implies that

$$g_u\left(t\right)\to +\infty \mathrm{as}t\to 0,$$

$$g_u\left(t\right)\to -\infty \mathrm{as}t\to +\infty ,$$

$$g_u^{\prime }\left(t\right)<0$$

and then there exists a unique $t_0>0$ such that

$$g_u\left(t_0\right)=\lambda {\int }_{{\mathbb{R}}^3}{\left|u\right|}^q\ln \left|u\right|dx$$

and

$$g_u^{\prime }\left(t_0\right)<0.$$

By (17) and (18), $t_{0}u\in {\mathcal{M}}^{-}$ for $\lambda >0$. □

Lemma 7.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $\lambda >0$ and $p<p\theta <q<p_s^{*}$, then

$$m_{\lambda }^{-}=\underset{{\mathcal{M}}^{-}}{\inf }I\left(u\right)=\underset{\mathcal{M}}{\inf }I\left(u\right)>0.$$

Proof. 

For all $u\in \mathcal{M}$, by (7) and (8), we get that

$$\langle {\Phi }^{\prime }\left(u\right),u\rangle ={(p-q)a\parallel u\parallel }_X^p+{(p\theta -q)b\parallel u\parallel }_X^{p\theta }-\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx<0,$$

then

$${\mathcal{M}}^{-}=\mathcal{M}.$$

If $\lambda >0$ and $u\in {\mathcal{M}}^{-},$ by (8) and Lemma 3 (i), there exists a constant $C_{\kappa }>0$ such that

$${\parallel u\parallel }_X>C_{\kappa }>0.$$

Further, from (7),

$$\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q{dx>(p-q)a\parallel u\parallel }_X^p+(p\theta -q)b{\parallel u\parallel }_X^{p\theta },$$

then

$$\begin{array}{l} I\left(u\right)-{\displaystyle \frac{1}{q}}\langle I^{\prime }\left(u\right),u\rangle \\ ={\displaystyle \frac{q-p}{pq}}{a\parallel u\parallel }_X^p+{\displaystyle \frac{q-p\theta }{pq\theta }}{b\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{1}{q^2}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\\ >{\displaystyle \frac{b{(q-p\theta )}^2}{p\theta q^2}}{\parallel u\parallel }_X^{p\theta }\\ >{\displaystyle \frac{b{(q-p\theta )}^2}{p\theta q^2}}{C}_{\kappa }^{p\theta }\\ >0.\end{array}$$

Thus, $m_{\lambda }^{-}={\inf }_{{\mathcal{M}}^{-}}I\left(u\right)={\inf }_{\mathcal{M}}I\left(u\right)>0.$ □

Proof of Theorem 1.

Let $\lambda >0$ and $p<p\theta <q<p_s^{*}$. According to Proposition $3.3$ and Lemma $5.2$ in [19], the Ekeland variational principle [20], Lemmas 4 and 9 (i), there exists a minimizing sequence $\left\{u_n\right\}\subset {\mathcal{M}}^{-}$ such that

$$I\left(u_n\right)=m_{\lambda }^{-}+o\left(1\right),I^{\prime }\left(u_n\right)=o\left(1\right)\mathrm{in}{X}^{*}.$$

By Lemma 5, there exists $u_0\in {\mathcal{M}}^{-}$ such that

$$u_n\to u_0\mathrm{in}X,I\left(u_0\right)=m_{\lambda }^{-}.$$

By Lemmas 2 and 7, problem (1) has a least energy solution, with a positive corresponding energy value. □

4. $\mathit{\lambda }<\mathbf{0}$: The Existence of Multiple Solutions

In this section, let $\lambda <0$; we obtain that Equation (1) has at least two solutions because the nonlinearity ${\lambda \left|u\right|}^{q-1}\ln \left|u\right|$ is both locally superlinear and locally sublinear. Firstly, consider the following fractional p-Laplacian Schrödinger equations under the assumption of $\left(V_1\right)$:

$$\left\{\begin{array}{l}{(-\Delta )}_p^{s}u+{V\left(x\right)\left|u\right|}^{p-2}u={\left|u\right|}^{q-2}u,\mathrm{in}{\mathbb{R}}^N,\\ u\in W^{s,p}\left({\mathbb{R}}^N\right),\end{array}\right.$$

(19)

where $0<s<1$, $ps<N$ and $1<p<q<p_s^{*}.$ Define

$$J\left(u\right)={\displaystyle \frac{1}{p}}{\parallel u\parallel }_X^p-{\displaystyle \frac{1}{q}}{\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx$$

and

$$\mathcal{N}=\{u\in X\setminus \left\{0\right\}|\langle J^{\prime }\left(u\right),u\rangle =\parallel u{\parallel }_X^p-{\int }_{{\mathbb{R}}^N}|u{|}^{q}dx=0\}.$$

From [6,15], there exists $\overline{u}\in \mathcal{N}$ such that

$$m=\underset{u\in \mathcal{N}}{\inf }J\left(u\right)=J\left(\overline{u}\right),$$

that is

$$m=J\left(\overline{u}\right)={\displaystyle \frac{1}{p}}\parallel \overline{u}{\parallel }_X^p-{\displaystyle \frac{1}{q}}{\int }_{{\mathbb{R}}^N}|\overline{u}{|}^{q}dx=\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}}\right){\parallel \overline{u}\parallel }_X^p;$$

hence,

$$\begin{array}{c}\hfill \parallel \overline{u}{\parallel }_X^p={\int }_{{\mathbb{R}}^N}{\left|\overline{u}\right|}^{q}dx={\displaystyle \frac{mpq}{q-p}}.\end{array}$$

(20)

To prove ${\mathcal{M}}^{\pm }\ne \varnothing$, we give a constant

$$\begin{array}{c}\hfill \alpha =\exp \left({\displaystyle \frac{S_{p_s^{*}}^{p_s^{*}}{\left(\frac{mpq}{q-p}\right)}^{\frac{q-p}{p}}}{e(p_s^{*}-q)}}\right)>1\end{array}$$

(21)

and the following Lemma 8. For the convenience of calculation, we assume that $\delta =1/e\alpha$ in the following.

Lemma 8.

Let $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}(\delta =1/e\alpha )$, then

(i) 

${\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx={\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}};$

(ii) 

${\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx\le {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \left(\alpha \delta \right)<0.$

Proof. 

(i) Let $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}$; then by (20); we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx\hfill \\ & & ={\displaystyle \frac{{\delta }^q}{\parallel \overline{u}{\parallel }_X^q}}{\int }_{{\mathbb{R}}^N}|\overline{u}{|}^{q}dx\hfill \\ & & ={\displaystyle \frac{{\delta }^q}{\parallel \overline{u}{\parallel }_X^q}}\parallel \overline{u}{\parallel }_X^p\hfill \\ & & ={\delta }^q{\parallel \overline{u}\parallel }_X^{p-q}\hfill \\ & & ={\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}.\hfill \end{array}$$

(22)

(ii) Let $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}$; we have that

$$\parallel \overline{v}{\parallel }_X^q={\displaystyle \frac{{\delta }^q{\parallel \overline{u}\parallel }_X^q}{\parallel \overline{u}{\parallel }_X^q}}={\delta }^q.$$

It follows from (22), the Sobolev embedding theorem, and the following inequality

$$\ln \xi \le {\displaystyle \frac{1}{e(p_s^{*}-q)}}{\xi }^{p_s^{*}-q}\mathrm{for}\xi >0$$

that

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx\hfill \\ & & ={\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|\delta \overline{u}{|}^q}{\parallel \overline{u}{\parallel }_X^q}}\ln {\displaystyle \frac{|\delta \overline{u}|}{\parallel \overline{u}{\parallel }_X}}dx\hfill \\ & & ={\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^q\ln \delta dx+{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{\delta }^q{\left|\overline{u}\right|}^q}{\parallel \overline{u}{\parallel }_X^q}}\ln {\displaystyle \frac{|\overline{u}|}{\parallel \overline{u}{\parallel }_X}}dx\hfill \\ & & \le \parallel \overline{u}{\parallel }_X^{p-q}{\delta }^q\ln \delta +{\displaystyle \frac{1}{e(p_s^{*}-q)}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{\delta }^q{\left|\overline{u}\right|}^q}{\parallel \overline{u}{\parallel }_X^q}}{\left({\displaystyle \frac{|\overline{u}|}{\parallel \overline{u}{\parallel }_X}}\right)}^{p_s^{*}-q}dx\hfill \\ & & ={\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \delta +{\displaystyle \frac{{\delta }^q}{e(p_s^{*}-q)}}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|\overline{u}{|}^{p_s^{*}}}{\parallel \overline{u}{\parallel }_X^{p_s^{*}}}}dx\hfill \\ & & \le {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \delta +{\displaystyle \frac{{\delta }^q}{e(p_s^{*}-q)}}{S}_{p_s^{*}}^{p_s^{*}}\hfill \\ & & ={\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \left(\alpha \delta \right),\hfill \end{array}$$

where $\alpha =\exp \left({\displaystyle \frac{S_{p_s^{*}}^{p_s^{*}}{\left(\frac{mpq}{q-p}\right)}^{\frac{q-p}{p}}}{e(p_s^{*}-q)}}\right)$ is given in (21).

Given that $0<\alpha \delta <1$, we have ${\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx\le {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \left(\alpha \delta \right)<0.$ □

Lemma 9.

Let ${\lambda }^{*}>0$ be given by (5) and $\delta =1/e\alpha$. Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $p<q<p_s^{*}$, $\lambda <-{\lambda }^{*}$ and $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}\in X\setminus \left\{0\right\}$, there exist $0<t_{-}<t_{\min }<t_{+}$ such that

(i) 

$t_{-}\overline{v}\in {\mathcal{M}}^{-}\mathrm{and}{t}_{+}\overline{v}\in {\mathcal{M}}^{+}.$ Moreover, $0<\parallel t_{-}\overline{v}{\parallel }_X<\parallel t_{\min }\overline{v}{\parallel }_X<{\parallel t_{+}\overline{v}\parallel }_X;$

(ii) 

$t_{+}>1$.

Proof. 

(i) For $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}\in X\setminus \left\{0\right\}$. Let

$$\begin{array}{c}\hfill g_{\overline{v}}\left(t\right)=t^{p-q}a\parallel \overline{v}{\parallel }_X^p+t^{p\theta -q}b\parallel \overline{v}{\parallel }_X^{p\theta }-\lambda \ln t{\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx,t>0,\end{array}$$

(23)

then

$$g_{\overline{v}}^{\prime }\left(t\right)=(p-q)t^{p-q-1}a\parallel \overline{v}{\parallel }_X^p+(p\theta -q)t^{p\theta -q-1}b\parallel \overline{v}{\parallel }_X^{p\theta }-\lambda t^{-1}{\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx.$$

Clearly,

$$t^{q+1}{g}_{\overline{v}}^{\prime }\left(t\right)=\langle {\Phi }^{\prime }\left(t\overline{v}\right),t\overline{v}\rangle$$

and

$$\begin{array}{c}\hfill t\overline{v}\in \mathcal{M}\iff g_{\overline{v}}\left(t\right)-\lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx=0.\end{array}$$

(24)

To prove $t\overline{v}\in {\mathcal{M}}^{+}$ (or $t\overline{v}\in {\mathcal{M}}^{-})$, we only need to prove

$$\begin{array}{c}\hfill g_{\overline{v}}\left(t\right)-\lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx=0\mathrm{and}{g}_{\overline{v}}^{\prime }\left(t\right)>0(\mathrm{or}{g}_{\overline{v}}^{\prime }\left(t\right)<0).\end{array}$$

(25)

If $\lambda <0$, $p<q<p_s^{*}$ and $\delta =1/e\alpha$. By $1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$ be given in $\left(M_1\right)$ and (23), we get that

$$g_{\overline{v}}\left(t\right)\to +\infty \mathrm{as}t\to 0$$

and

$$g_{\overline{v}}\left(t\right)\to +\infty \mathrm{as}t\to +\infty .$$

Hence, $g_{\overline{v}}\left(t\right)$ has the unique minimum point $t_{min}$ on $(0,+\infty )$ and

$$\parallel \overline{v}{\parallel }_X=\delta >0.$$

Consider the following auxiliary function

$$\begin{array}{c}\hfill {\Pi }_{\overline{v}}\left(t\right)=t^{p-q}\left(a\parallel \overline{v}{\parallel }_X^p+b{\parallel \overline{v}\parallel }_X^{p\theta }\right)-\lambda \ln t{\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx,t>0.\end{array}$$

(26)

By standard calculations and (22), we obtain that ${\Pi }_{\overline{v}}\left(t\right)$ has the unique minimum point $\overline{t}$ on $(0,+\infty )$, that is,

$$\begin{array}{ll}\hfill \overline{t}& ={\left({\displaystyle \frac{(q-p)(a\parallel \overline{v}{\parallel }_X^p+b\parallel \overline{v}{\parallel }_X^{p\theta })}{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}}\right)}^{{\displaystyle \frac{1}{q-p}}}\\ & ={\left({\displaystyle \frac{(q-p)(a\parallel \overline{v}{\parallel }_X^p+b\parallel \overline{v}{\parallel }_X^{p\theta })}{-\lambda {\delta }^q{\parallel \overline{u}\parallel }_X^{p-q}}}\right)}^{{\displaystyle \frac{1}{q-p}}}\\ & ={\left({\displaystyle \frac{(q-p)\left(a{\delta }^{p-q}+b{\delta }^{p\theta -q}\right)}{-\lambda }}\right)}^{{\displaystyle \frac{1}{q-p}}}{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{1}{p}}}\\ & <{\left({\displaystyle \frac{1}{e}}\right)}^{{\displaystyle \frac{1}{q-p}}}\\ & <1,\end{array}$$

for $\lambda <-{\lambda }^{*}<0$, where ${\lambda }^{*}$ is given by (5), then

$${\overline{t}}^{q-p}=\left({\displaystyle \frac{(q-p)(a\parallel \overline{v}{\parallel }_X^p+b\parallel \overline{v}{\parallel }_X^{p\theta })}{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}}\right)<{\displaystyle \frac{1}{e}}$$

and

$$\begin{array}{ccc}\hfill & & \underset{t>0}{\min }{\Pi }_{\overline{v}}\left(t\right)={\Pi }_{\overline{v}}\left(\overline{t}\right)\hfill \\ & & ={\Pi }_{\overline{v}}\left({\left({\displaystyle \frac{(q-p)(a\parallel \overline{v}{\parallel }_X^p+b\parallel \overline{v}{\parallel }_X^{p\theta })}{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}}\right)}^{{\displaystyle \frac{1}{q-p}}}\right)\hfill \\ & & ={\displaystyle \frac{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}{q-p}}\ln e+{\displaystyle \frac{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}{q-p}}\ln {\displaystyle \frac{(q-p)\left(a\parallel \overline{v}{\parallel }_X^p+b{\parallel \overline{v}\parallel }_X^{p\theta }\right)}{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}}\hfill \\ & & ={\displaystyle \frac{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}{q-p}}\ln {\displaystyle \frac{e(q-p)\left(a\parallel \overline{v}{\parallel }_X^p+b{\parallel \overline{v}\parallel }_X^{p\theta }\right)}{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}}\hfill \\ & & <{\displaystyle \frac{-\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx}{q-p}}\ln 1\hfill \\ & & =0.\hfill \end{array}$$

(27)

By (23), (26), and $\overline{t}<1$, we have that

$$\begin{array}{c}\hfill g_{\overline{v}}\left(\overline{t}\right)<{\Pi }_{\overline{v}}\left(\overline{t}\right).\end{array}$$

(28)

From (27) and (28), we get that

$$\begin{array}{c}\hfill \underset{t>0}{\min }{g}_{\overline{v}}\left(t\right)=g_{\overline{v}}\left(t_{\min }\right)\le g_{\overline{v}}\left(\overline{t}\right)<{\Pi }_{\overline{v}}\left(\overline{t}\right)<0.\end{array}$$

(29)

Since $\lambda <-{\lambda }^{*}<0$, Lemma 8 (ii) implies that

$$\begin{array}{c}\hfill \lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx>0.\end{array}$$

(30)

By (29) and (30), there exist $0<t_{-}<t_1<t_{\min }<t_2<t_{+}$ such that

$$g_{\overline{v}}\left(t_1\right)=g_{\overline{v}}\left(t_2\right)=0,$$

$$g_{\overline{v}}\left(t_{-}\right)=\lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx>0,g_{\overline{v}}^{\prime }\left(t_{-}\right)<0$$

and

$$\begin{array}{c}\hfill g_{\overline{v}}\left(t_{+}\right)=\lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx>0,g_{\overline{v}}^{\prime }\left(t_{+}\right)>0.\end{array}$$

(31)

By (24) and (25), we get that

$$t_{-}\overline{v}\in {\mathcal{M}}^{-}\mathrm{and}{t}_{+}\overline{v}\in {\mathcal{M}}^{+}.$$

Moreover,

$$0<\parallel t_{-}\overline{v}{\parallel }_X<\parallel t_1\overline{v}{\parallel }_X<\parallel t_{\min }\overline{v}{\parallel }_X<\parallel t_2\overline{v}{\parallel }_X<{\parallel t_{+}\overline{v}\parallel }_X.$$

(ii) If $\lambda <0$, $p<q<p_s^{*}$ and $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}$. Choosing $\delta =1/e\alpha$, from Lemma 8, (23), and (31), we have that

$$g_{\overline{v}}\left(1\right)=a\parallel \overline{v}{\parallel }_X^p+b{\parallel \overline{v}\parallel }_X^{p\theta }=a{\delta }^p+b{\delta }^{p\theta }$$

and

$$g_{\overline{v}}\left(t_{+}\right)=\lambda {\int }_{{\mathbb{R}}^N}|\overline{v}{|}^q\ln \left|\overline{v}\right|dx\ge \lambda {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\ln \left(\alpha \delta \right)>-\lambda {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}.$$

Then

$$0<g_{\overline{v}}\left(1\right)<g_{\overline{v}}\left(t_{+}\right)\mathrm{for}\lambda <-{\lambda }^{*}<0.$$

According to (31) again, $g_{\overline{v}}^{\prime }\left(t_{+}\right)>0$ implies that $t_{+}>1$ for $\lambda <-{\lambda }^{*}<0.$ □

By Lemma 9, we define

$${\mathcal{M}}_{\delta }^{-}={\{u\in \mathcal{M}:\parallel u\parallel }_X<D_1\}$$

and

$${\mathcal{M}}_{\delta }^{+}={\{u\in \mathcal{M}:\parallel u\parallel }_X>D_2\},$$

where $D_1={\parallel t_1\overline{v}\parallel }_X=t_1\delta$ and $D_2={\parallel t_2\overline{v}\parallel }_X=t_2\delta$, and then

$$D_1<t_{\min }\delta <D_2.$$

From this, the following Lemma 10 is obtained.

Lemma 10.

Let ${\lambda }^{*}>0$ be given by (5). Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. Then, if $\lambda <-{\lambda }^{*}$ and $p<q<p_s^{*}$, then

$${\mathcal{M}}_{\delta }^{+}\cap {\mathcal{M}}_{\delta }^{-}=\varnothing .$$

Proof. 

The proof of can be obtained by Lemma 9, so we omit it. □

Lemma 11.

Assume that $\left(V_1\right)$ and $\left(M_1\right)$ hold. If $\lambda <-{\lambda }^{*}<0$ and $p<q<p_s^{*}$, then

(i) 

$m^{+}={\inf }_{{\mathcal{M}}_{\delta }^{+}}I\left(u\right)<0;$

(ii) 

$m^{-}={\inf }_{{\mathcal{M}}_{\delta }^{-}}I\left(u\right)>0$.

Proof. 

(i) If $\lambda <-{\lambda }^{*}<0$, $\overline{v}={\displaystyle \frac{\delta \overline{u}}{\parallel \overline{u}{\parallel }_X}}$ and $1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$ be given in $\left(M_1\right)$. By Lemma 9, for $\delta \in (0,{\displaystyle \frac{1}{e\alpha }})$, there exists $t_{+}>1$ such that

$$t_{+}\overline{v}\in {\mathcal{M}}_{\delta }^{+}.$$

Either when $p<p\theta <q\mathrm{or}p<q<p\theta$, we have that

$$\begin{array}{l} I\left(t_{+}\overline{v}\right)\\ =I\left(t_{+}\overline{v}\right)-{\displaystyle \frac{1}{q}}\langle I^{\prime }\left(t_{+}\overline{v}\right),t_{+}\overline{v}\rangle \\ ={\displaystyle \frac{q-p}{pq}}a{t}_{+}^p\parallel \overline{v}{\parallel }_X^p+{\displaystyle \frac{q-p\theta }{pq\theta }}b{t}_{+}^{p\theta }\parallel \overline{v}{\parallel }_X^{p\theta }+{\displaystyle \frac{1}{q^2}}{t}_{+}^q\lambda {\int }_{{\mathbb{R}}^N}{\left|\overline{v}\right|}^{q}dx\\ <t_{+}^q\left({\displaystyle \frac{q-p}{pq}}a{\delta }^p+{\displaystyle \frac{q-p}{pq\theta }}b{\delta }^{p\theta }+{\displaystyle \frac{1}{q^2}}\lambda {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\right)\\ <t_{+}^q\left({\displaystyle \frac{q-p}{pq}}\left(a{\delta }^p+b{\delta }^{p\theta }\right)+{\displaystyle \frac{1}{q^2}}\lambda {\delta }^q{\left({\displaystyle \frac{mpq}{q-p}}\right)}^{{\displaystyle \frac{p-q}{p}}}\right)\\ <0\end{array}$$

for $\lambda <-{\lambda }^{*}<0$. Thus, $m^{+}={\inf }_{{\mathcal{M}}_{\delta }^{+}}I\left(u\right)<0.$

(ii) If $\lambda <0$ and $u\in {\mathcal{M}}_{\delta }^{-},$ by (8) and Lemma 3 (i), there exists a constant $C_{\kappa }>0$ such that

$${\parallel u\parallel }_X>C_{\kappa }>0.$$

Further, from (7),

$$\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^q{dx>(p-q)a\parallel u\parallel }_X^p+(p\theta -q)b{\parallel u\parallel }_X^{p\theta },$$

then

$$\begin{array}{l} I\left(u\right)-{\displaystyle \frac{1}{q}}\langle I^{\prime }\left(u\right),u\rangle \\ ={\displaystyle \frac{q-p}{pq}}{a\parallel u\parallel }_X^p+{\displaystyle \frac{q-p\theta }{pq\theta }}{b\parallel u\parallel }_X^{p\theta }+{\displaystyle \frac{1}{q^2}}\lambda {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{q}dx\\ >{\displaystyle \frac{a{(q-p)}^2}{p{q}^2}}{\parallel u\parallel }_X^p+{\displaystyle \frac{b{(q-p\theta )}^2}{p\theta q^2}}{\parallel u\parallel }_X^{p\theta }\\ >{\displaystyle \frac{b{(q-p\theta )}^2}{p\theta q^2}}{C}_{\kappa }^{p\theta }\\ >0.\end{array}$$

Thus, $m^{-}={\inf }_{{\mathcal{M}}_{\delta }^{-}}I\left(u\right)>0.$ □

Proof of Theorem 2.

Let By Lemma 5, there exists $\lambda <-{\lambda }^{*}<0$ and $p<q<p_s^{*}$. Similarly to the proof of Theorem 1, according to Proposition 3.3 and Lemma 5.2 in [19], the Ekeland variational principle [20] and Lemma 9 (ii), we determine that there exist two minimizing sequences $\left\{u_n^{\pm }\right\}\subset {\mathcal{M}}_{\delta }^{\pm }$ such that

$$I\left(u_n^{\pm }\right)=m^{\pm }+o\left(1\right),I^{\prime }\left(u_n^{\pm }\right)=o\left(1\right)\mathrm{in}{X}^{*}.$$

By Lemma 5, there exist two minimizers $u^{\pm }\in {\mathcal{M}}_{\delta }^{\pm }$ such that

$$u_n^{\pm }\to u^{\pm }\mathrm{in}X,I\left(u^{\pm }\right)=m^{\pm }.$$

Lemmas 9 (i) and 10 imply that $0<\parallel u^{-}{\parallel }_X<{\parallel u^{+}\parallel }_X$. By Lemmas 2 and 11, problem (1) has at least two nontrivial solutions, $u^{-}\in X$ and $u^{+}\in X$, with positive and negative corresponding energy values, respectively. □