Infinitely many solutions for a class of sublinear fractional Schrödinger equations with indefinite potentials

Abstract

In this paper, we consider the following sublinear fractional Schrödinger equation:

$$ (-\Delta)^{s}u + V(x)u= K(x) \vert u \vert ^{p-1}u,\quad x\in \mathbb{R}^{N}, $$

where $s, p\in(0,1)$, $N>2s$, $(-\Delta)^{s}$ is a fractional Laplacian operator, and K, V both change sign in $\mathbb{R}^{N}$. We prove that the problem has infinitely many solutions under appropriate assumptions on K, V. The tool used in this paper is the symmetric mountain pass theorem.

1 Introduction and main result

In this paper, we consider the following sublinear fractional Schrödinger equation:

$$ (-\Delta)^{s}u + V(x)u= K(x) \vert u \vert ^{p-1}u,\quad x\in\mathbb{R}^{N}, $$

(1.1)

where $s, p\in(0,1)$, $N>2s$, $(-\Delta)^{s}$ is a fractional Laplacian operator, K, V both change sign in $\mathbb{R}^{N}$ and satisfy some conditions specified below.

Problem (1.1) gives the following nonlinear field equation:

$$ i\frac{\partial\varPsi}{\partial t}=(-\Delta)^{s} \varPsi+ (1+E)\varPsi- K(x) \vert \varPsi \vert ^{p-1}\varPsi,\quad x\in\mathbb{R}^{N}, t\in \mathbb{R^{+}}. $$

(1.2)

The nonlinear field Eq. (1.2) reflects the stable diffusion process of Lévy particles in random field. Later, people found that this stable diffusion of Lévy process has also a very important application in the mechanical system, flame propagation, chemical reactions in the liquid, and the anomalous diffusion of physics in the plasma. For more details, readers can refer to [5, 25, 26, 45] and the references therein.

Problem (1.1) involves the fractional Laplacian $(-\bigtriangleup )^{s}$, which is a nonlocal operator. After this question was raised, it immediately aroused the interest of mathematicians (see [1, 4, 6–14, 16–22, 24, 27–29, 31, 33–44, 46–55] and the references therein).

For fractional equations on the whole space $\mathbb{R}^{N}$, the main difficulty one may face is that the Sobolev embedding $H^{s}(\mathbb {R}^{N})\hookrightarrow L^{q}(\mathbb{R}^{N})$ is not compact for $q\in [2, 2^{\ast}_{s})$. To overcome this difficulty, some authors [8, 10, 24, 31, 38, 50] considered fractional equations with the potential V satisfying the following conditions:

$(V)$:

$V\in C(\mathbb{R}^{N}, \mathbb{R})$, $\inf_{x\in\mathbb {R}^{N}}V(x)\geq V_{0}>0$ and, for each $M>0$, $\operatorname{meas}\{x\in\mathbb {R}^{N}: V(x)\leq M\}<\infty$, where $V_{0}$ is a constant and meas denotes Lebesgue measure in $\mathbb{R}^{N}$.

Due to condition $(V)$, the subspace of $H^{s}(\mathbb{R}^{N})$ embeds compactly into $L^{q}(\mathbb{R}^{N})$ for $q\in[2, 2^{\ast}_{s})$, which is crucial in their paper. In fact, condition $(V)$ is certain coercive condition. In the case of coercive condition $\lim_{|x|\rightarrow+\infty}V(x)=+\infty$, some authors, for example [12, 33], considered fractional equations on the whole space $\mathbb{R}^{N}$.

To overcome the difficulties caused by the lack of compactness, on the other hand, some authors restricted the energy functional to a subspace for $H^{s}(\mathbb{R}^{N})$ of radially symmetric functions, which embeds compactly into $L^{s}(\mathbb{R}^{N})$, for example, [9, 21, 34, 44, 54].

However, in this paper, we do not need some conditions like $(V)$ or radially symmetric. That is, our paper does not use any compact embedding on the whole space $\mathbb{R}^{N}$.

It is worth noting that, for fractional equations on the whole space $\mathbb{R}^{N}$, most results need condition $V(x)\geq0$ (see [1, 8–10, 12, 13, 16, 18, 20–22, 24, 28, 33, 34, 36–38, 44, 50, 52–54], in which some results were obtained in case of $V(x)=1$ [16, 18, 21, 28, 44]). To the best of our knowledge, there are few results on the existence of solutions for fractional equations with a sign-changing potential except [11, 51]. In fact, replaced $\inf_{x\in \mathbb{R}^{N}}V(x)\geq V_{0}>0$ with $\inf_{x\in\mathbb {R}^{N}}V(x)>-\infty$, condition similar to $(V)$ is needed in [11]. In [51], Xu, Wei, and Dong considered the following p-Laplacian equation with positive nonlinearity:

$$\begin{aligned} (-\Delta)_{p}^{s} u+V(x) \vert u \vert ^{p-2}u- \lambda \vert u \vert ^{p-2}u=f(x,u)+g(x) \vert u \vert ^{q-2}u,\quad x\in\mathbb{R}^{N}, \end{aligned}$$

where $N, p\geq2$, $s\in(0,1)$, λ is a parameter, $(-\Delta )_{p}^{s}$ is the fractional p-Laplacian, and $f: \mathbb {R}^{N}\times\mathbb{R} \rightarrow\mathbb{R}$ is a Carathéodory function. In the case of $\lambda=0$, they obtained the existence of a nontrivial solution to this equation. Furthermore, they proved that this equation has infinitely many nontrivial solutions when $\lambda\leq0$ or $\lambda>0$ is small enough.

In this article, we are interested in the existence of infinitely many solutions for problem (1.1) with potential function $V(x)$ changing sign in $\mathbb{R}^{N}$. Moreover, nonlinearity can be allowed to change sign. To state our main result, we assume the following:

$(V_{1})$:

$V\in L^{\infty}(\mathbb{R}^{N})$ and there exist $\alpha, R_{0}>0$ such that

$$V(x)\geq\alpha,\quad \forall \vert x \vert \geq R_{0}. $$

$(V_{2})$:

$\|V^{-}\|_{\frac{N}{2s}}<\frac{1}{S}$, where $V^{\pm }(x)=\max\{\pm V(x),0\}$ and S is the constant of Sobolev:

$$\Vert u \Vert ^{2}_{2_{s}^{\ast}}\leq S \Vert u \Vert ^{2}_{H_{0}^{s}(\mathbb{R}^{N})},\quad \forall u\in H^{s}\bigl( \mathbb{R}^{N}\bigr), \text{where } 2_{s}^{\ast}= \frac{2N}{N-2s}. $$

$(K)$:

$K\in L^{\infty}(\mathbb{R}^{N})$ and there exist $\beta>0$, $R_{1}>R_{2}>0$, $y_{0}=(y_{1},\ldots,y_{N})\in\mathbb{R}^{N}$ such that

$$K(x)\leq-\beta,\quad \forall \vert x \vert >R_{1};\qquad K(x)>0, \quad\forall x \in B(y_{0},R_{2})\subset B(0,R_{1}). $$

Our main result of this paper can be stated as follows.

Theorem 1.1

Assume$(V_{1})$–$(V_{2})$and$(K)$hold. Then problem (1.1) possesses infinitely many nontrivial solutions.

Remark 1.1

The ideas in this article come from the paper [3], where Schrödinger equations were considered. However, our proof is nontrivial since we present a simplified proof for the PS condition by comparing to that in [3]. In fact, the PS condition was proved in [3] by concentration compactness principle. It is noticed that the PS condition plays important role in the proof of the main results in [3].

2 Notations and preliminaries

In this paper, we use the following notations. Let

$$\Vert u \Vert _{q}= \biggl( \int_{{\mathbb {R}}^{N}} \vert u \vert ^{q}\,dx \biggr)^{\frac{1}{q}},\quad 1\leq q< +\infty. $$

Let E be a Banach space and $\varphi:E\rightarrow {\mathbb {R}}$ be a functional of class $C^{1}$. The Fréchet derivative of φ at u, $\varphi'(u)$ is an element of the dual space $E^{\ast}$, and we denote $\varphi'(u)$ evaluated at $v\in E$ by $\langle\varphi '(u),v\rangle$.

Let $s\in(0,1)$, the fractional Sobolev space $H^{s}({\mathbb {R}}^{N})$ is defined by

$$H^{s}\bigl({\mathbb {R}}^{N}\bigr)=\biggl\{ u\in L^{2}\bigl( {\mathbb {R}}^{N}\bigr):\frac{ \vert u(x)-u(y) \vert }{ \vert x-y \vert ^{\frac {N}{2}+s}}\in L^{2}\bigl( {\mathbb {R}}^{N}\times {\mathbb {R}}^{N}\bigr)\biggr\} $$

and endowed with the natural norm

$$\Vert u \Vert _{H^{s}({\mathbb {R}}^{N})}= \biggl( \int_{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx+ \int _{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}}\frac { \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{N+2s}}\,dx\,dy \biggr)^{\frac{1}{2}}, $$

here

$$[u]_{H^{s}(\mathbb{R}^{N})}= \biggl( \int_{\mathbb{R}^{N}} \int_{\mathbb {R}^{N}}\frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{N+2s}}\,dx\,dy \biggr)^{\frac{1}{2}} $$

is the so-called Gagliardo (semi) norm of u.

Using Fourier transform, the space $H^{s}(\mathbb{R}^{N})$ can also be defined by

$$H^{s}\bigl(\mathbb{R}^{N}\bigr)=\biggl\{ u\in L^{2}\bigl(\mathbb{R}^{N}\bigr): \int_{\mathbb {R}^{N}}\bigl(1+ \vert \xi \vert ^{2s}\bigr) \vert \mathscr{F} u \vert ^{2}\,d\xi< +\infty\biggr\} , $$

where $\mathscr{F} u$ denotes the Fourier transform of u.

Let ℓ be the Schwartz space of rapidly decreasing $C^{\infty}$ function on $\mathbb{R}^{N}$, $u\in\ell$, one has

$$(-\bigtriangleup)^{s}u(x)=C(N,s)\textit{P.V.} \int_{\mathbb{R}^{N}}\frac {u(x)-u(y)}{ \vert x-y \vert ^{N+2s}}\,dy, $$

the symbol P.V. stands for the Cauchy value, and $C(N,s)$ is a constant dependent only on the space dimension N and the order s.

From the results of [15], we have

$$(-\bigtriangleup)^{s}u=\mathscr{F} ^{-1}\bigl( \vert \xi \vert ^{2s}(\mathscr{F}u)\bigr) \quad\text{for any } \xi\in \mathbb{R}^{N}. $$

Then, by Proposition 3.4 and Proposition 3.6 of [15], we have

$$[u]^{2}_{H^{s}}=\frac{2}{C(N,s)} \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi=\frac{2}{C(N,s)} \bigl\Vert (- \bigtriangleup )^{\frac{s}{2}}u \bigr\Vert ^{2}_{2}. $$

From the above facts, the norms on $H^{s}(\mathbb{R}^{N})$ defined as follows

$$\begin{gathered} u\mapsto \biggl( \Vert u \Vert ^{2}_{2}+ \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr {F} u \vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}, \\ u\mapsto \bigl( \Vert u \Vert ^{2}_{2}+ \bigl\Vert (-\bigtriangleup)^{\frac{s}{2}}u \bigr\Vert ^{2}_{2} \bigr)^{\frac{1}{2}}, \\ u\mapsto \Vert u \Vert _{H^{s}(\mathbb{R}^{N})}\end{gathered} $$

are all equivalent.

Lemma 2.1

([15, 30, 34])

Let$0< s<1$such that$2s< N$. Then there exists$C=C(n,s)$such that

$$\Vert u \Vert _{2_{s}^{\ast}}\leq C \Vert u \Vert _{H^{s}(\mathbb{R}^{N})} $$

for every$u\in H^{s}(\mathbb{R}^{N})$. Moreover, the embedding$H^{s}(\mathbb{R}^{N})\subset L^{p}(\mathbb{R}^{N})$is continuous for any$p\in[2,2_{s}^{\ast}]$and locally compact whenever$p\in [2,2_{s}^{\ast})$.

Let the homogeneous Sobolev space

$$H_{0}^{s}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u\in L^{2_{s}^{\ast}}\bigl(\mathbb{R}^{N}\bigr): \vert \xi \vert ^{s}\mathscr{F} u\in L^{2}\bigl(\mathbb{R}^{N} \bigr)\bigr\} . $$

This space can be equivalently defined as the completion of $C_{0}^{\infty}(\mathbb{R}^{N})$ under the norm

$$\Vert u \Vert _{0}^{2}\triangleq \Vert u \Vert ^{2}_{H_{0}^{s}(\mathbb{R}^{N})}\triangleq \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi. $$

The Sobolev space $E=H^{s}(\mathbb{R}^{N})\cap L^{p+1}(\mathbb{R}^{N})$ is endowed with the norm

$$\Vert u \Vert = \Vert u \Vert _{0}+ \Vert u \Vert _{p+1}. $$

Obviously, E is a reflexive Banach space.

The energy functional $\varphi:E\rightarrow {\mathbb {R}}$ corresponding to problem (1.1) is defined by

$$\varphi(u)=\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi+\frac{1}{2} \int_{\mathbb{R}^{N}}V(x)u^{2}\,dx-\frac {1}{p+1} \int_{\mathbb{R}^{N}}K(x) \vert u \vert ^{p+1}\,dx. $$

Under our conditions, $\varphi\in C^{1}(E)$ and its critical points are solutions of problem (1.1).

Definition 2.1

([32])

Let E be a Banach space and A be a subset of E. Set A is said to be symmetric if $u\in E$ implies $-u\in E$. For a closed symmetric set A which does not contain the origin, we define a genus $\gamma(A)$ of A by the smallest integer k such that there exists an odd continuous mapping from A to ${\mathbb {R}}^{k}\setminus\{0\}$. If there does not exist such k, we define $\gamma(A)=\infty$. We set $\gamma (\emptyset)=0$. Let $\varGamma_{k}$ denote the family of closed symmetric subsets A of E such that $0\notin A$ and $\gamma(A)\geq k$.

The following result is a version of the classical symmetric mountain pass theorem [2, 32]. For the proof, please see [23].

Theorem 2.1

([23])

LetEbe an infinite dimensional Banach space and$I\in C^{1}(E,{\mathbb {R}})$satisfy:

$(I_{1})$:

Iis even, bounded from below, $I(0)=0$, andIsatisfies the Palais–Smale condition.

$(I_{2})$:

For each$k\in {\mathbb {N}}$, there exists$A_{k}\in\varGamma_{k}$such that

$$\sup_{u\in A_{k}}I(u)< 0. $$

Then either of the following two conditions holds:

• (i) there exists a sequence${u_{k}}$such that$I'(u_{k})=0, I(u_{k})<0$and${u_{k}}$converges to zero; or

• (ii) there exist two sequences${u_{k}}$and${v_{k}}$such that$I'(u_{k})=0$, $I(u_{k})=0$, $u_{k}\neq0$, $\lim_{k\rightarrow+\infty }u_{k}=0$, $I'(v_{k})=0$, $I(v_{k})<0$, $\lim_{k\rightarrow+\infty }I(v_{k})=0$and${v_{k}}$converges to a non-zero limit.

3 Proof of Theorem 1.1

Lemma 3.1

Suppose that$(V_{1})$–$(V_{2})$and$(K)$hold. Then any PS sequence ofφis bounded inE.

Proof

Let $\{u_{n}\}\subset E$ be such that

$$\varphi(u_{n}) \text{ is bounded}\quad \text{and}\quad\varphi'(u_{n}) \rightarrow0 \quad\text{as } n\rightarrow\infty. $$

That is, there exists $C>0$ such that $\varphi(u_{n})\leq C$. So, according to Hölder’s inequality and Sobolev’s inequality, one has that

$$ \begin{aligned} C &\geq\varphi(u_{n})= \frac{1}{2} \int _{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u_{n} \vert ^{2}\,d\xi+\frac {1}{2} \int_{\mathbb{R}^{N}}V(x)u_{n}^{2}\,dx- \frac{1}{p+1} \int _{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p+1} \,dx \\ &\geq\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u_{n} \vert ^{2}\,d\xi-\frac{1}{2} \int_{\mathbb {R}^{N}}V^{-}(x)u_{n}^{2} \,dx-\frac{1}{p+1} \int_{\mathbb {R}^{N}}K^{+}(x) \vert u_{n} \vert ^{p+1}\,dx \\ &\geq\frac{1}{2} \Vert u_{n} \Vert ^{2}_{0}-\frac{1}{2} \biggl( \int_{\mathbb{R}^{N}} \bigl\vert V^{-} \bigr\vert ^{\frac{N}{2s}}\,dx \biggr)^{\frac {2s}{N}} \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert u_{n} \vert ^{2}\bigr)^{\frac{2_{s}^{\ast }}{2}}\,dx \biggr)^{\frac{2}{2_{s}^{\ast}}} \\ &\quad-\frac{1}{p+1} \int_{\mathbb{R}^{N}}K^{+}(x) \vert u_{n} \vert ^{p+1}\,dx \\ &\geq \biggl(\frac{1}{2}-\frac{S}{2} \bigl\Vert V^{-} \bigr\Vert _{\frac {N}{2s}} \biggr) \Vert u_{n} \Vert ^{2}_{0}-\frac{S^{\frac{p+1}{2}}}{p+1} \bigl\Vert K^{+} \bigr\Vert _{\frac{2_{s}^{\ast}}{2_{s}^{\ast}-(p+1)}} \Vert u_{n} \Vert _{0}^{p+1}.\end{aligned} $$

Since $0< p<1$, there exists $\eta>0$ such that

$$ \Vert u_{n} \Vert ^{2}_{0} \leq\eta,\quad \forall n\in {\mathbb {N}}. $$

(3.1)

On the other hand, we have that

$$ \begin{aligned} C +\frac{ \Vert u_{n} \Vert }{2}& \geq\varphi(u_{n})- \frac {1}{2}\bigl\langle \varphi'(u_{n}),u_{n} \bigr\rangle \\ &\geq \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr) \int _{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p+1} \,dx \\ &= \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr) \int _{\mathbb{R}^{N}}K^{+}(x) \vert u_{n} \vert ^{p+1}\,dx+ \biggl(\frac {1}{p+1}-\frac{1}{2} \biggr) \int_{\mathbb {R}^{N}}K^{-}(x) \vert u_{n} \vert ^{p+1}\,dx \\ &= \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr) \int _{\mathbb{R}^{N}} \bigl(K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert u_{n} \vert ^{p+1}\,dx \\ &\quad+ \biggl(\frac{1}{p+1}-\frac{1}{2} \biggr) \int_{\mathbb{R}^{N}} \bigl(K^{-}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert u_{n} \vert ^{p+1}\,dx,\end{aligned} $$

where $\|\cdot\|$ denotes the norm in E.

Thanks to $(K)$, we have that

$$ K^{+}(x)=0 \quad\text{for all } \vert x \vert > R_{1}. $$

Then, by $K\in L^{\infty}(\mathbb{R}^{N})$, we get

$$\begin{aligned} \int_{\mathbb{R}^{N}} \bigl\vert K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr\vert ^{\frac {2_{s}^{\ast}}{2_{s}^{\ast}-(p+1)}}\,dx = \int_{B(0,R_{1})} \bigl\vert K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr\vert ^{\frac{2_{s}^{\ast }}{2_{s}^{\ast}-(p+1)}}\,dx< \infty. \end{aligned}$$

Hence, by Hölder’s inequality and Sobolev’s inequality, we have that

$$\begin{aligned}& \int_{\mathbb{R}^{N}} \bigl(K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert u_{n} \vert ^{p+1}\,dx \\& \quad\leq \biggl( \int_{\mathbb{R}^{N}} \bigl(K^{+}(x)+\chi _{B(0,R_{1})}(x) \bigr)^{\frac{2_{s}^{\ast}}{2_{s}^{\ast}-(p+1)}}\, dx \biggr)^{\frac{2_{s}^{\ast}-(p+1)}{2_{s}^{\ast}}}\times \biggl( \int_{\mathbb{R}^{N}} \bigl( \vert u_{n} \vert ^{p+1} \bigr)^{\frac{2_{s}^{\ast }}{p+1}}\,dx \biggr)^{\frac{p+1}{2_{s}^{\ast}}} \\& \quad\leq S^{\frac{p+1}{2}} \bigl\Vert K^{+}+\chi_{B(0,R_{1})} \bigr\Vert _{\frac {2_{s}^{\ast}}{2_{s}^{\ast}-(p+1)}} \Vert u_{n} \Vert _{0}^{p+1}. \end{aligned}$$

(3.2)

Using $(K)$ again, we know that $K^{-}(x)\geq\beta$ for all $|x|> R_{1}$. Then we have that

$$ \int_{\mathbb{R}^{N}} \bigl(K^{-}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert u_{n} \vert ^{p+1}\,dx\geq \min(\beta,1) \Vert u_{n} \Vert _{p+1}^{p+1}. $$

(3.3)

According to (3.1), (3.2), and (3.3), there exists a constant $C_{1}>0$ such that

$$\Vert u_{n} \Vert ^{p+1}_{p+1}\leq C_{1}+C_{1} \Vert u_{n} \Vert _{p+1}\quad \text{for all } n\in {\mathbb {N}}. $$

Since $0< p<1$, there exists a constant $C_{2}>0$ such that

$$ \Vert u_{n} \Vert _{p+1}\leq C_{2},\quad \forall n\in {\mathbb {N}}. $$

(3.4)

Hence, it follows from (3.1) and (3.4) that $\{ u_{n}\}$ is bounded in E. □

Lemma 3.2

Suppose that$(V_{1})$–$(V_{2})$and$(K)$hold. Thenφsatisfies the PS condition onE.

Proof

Let $\{u_{n}\}\subset E$ be such that

$$\varphi(u_{n}) \text{ is bounded}\quad \text{and} \quad\varphi'(u_{n}) \rightarrow0 \quad\text{as } n\rightarrow\infty. $$

By Lemma 3.1, $\{u_{n}\}$ is bounded in E. Going if necessary to a subsequence, from Lemma 2.1 we can assume that

$$\begin{aligned} u_{n}\rightharpoonup u \text{ in } E;\qquad u_{n} \rightarrow u \text{ in } L^{q}_{\mathrm{loc}}\bigl(\mathbb{R}^{N} \bigr), \quad 2\leq q< 2_{s}^{\ast};\qquad u_{n}\rightarrow u \text{ a.e in } \mathbb{R}^{N}. \end{aligned}$$

(3.5)

So, $\forall\psi\in C^{\infty}_{0}(\mathbb{R}^{N})$, we have

$$\int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s}\mathscr{F} u_{n}\mathscr{F} \psi \,d\xi+ \int_{\mathbb{R}^{N}}V(x)u_{n}\psi\,dx\rightarrow \int_{\mathbb {R}^{N}} \vert \xi \vert ^{2s}\mathscr{F} u \mathscr{F} \psi \,d\xi+ \int_{\mathbb {R}^{N}}V(x)u\psi\,dx. $$

By $u_{n}\rightarrow u$ in $L^{p+1}(\operatorname{supp}(\psi))$ [15, 30] and Lebesgue’s dominated convergence theorem, one has that

$$\int_{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p-1}u_{n}\psi\,dx\rightarrow \int _{\mathbb{R}^{N}}K(x) \vert u \vert ^{p-1}u\psi\,dx. $$

Hence, we have

$$0=\lim_{n\rightarrow+\infty}\bigl\langle \varphi'(u_{n}), \psi\bigr\rangle =\bigl\langle \varphi'(u),\psi\bigr\rangle ,\quad \forall \psi\in C^{\infty }_{0}\bigl(\mathbb{R}^{N}\bigr). $$

Then

$$ \bigl\langle \varphi'(u),u\bigr\rangle =0. $$

Let $v_{n}=u_{n}-u$, then $u_{n}=v_{n}+u$, we have that

$$ \begin{aligned} \bigl\langle \varphi'(u_{n}),u_{n} \bigr\rangle &= \int_{\mathbb {R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u_{n} \vert ^{2}\,d\xi+ \int_{\mathbb {R}^{N}}V(x)u_{n}^{2}\,dx- \int_{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p+1} \,dx \\ &= \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \bigl( \vert \mathscr{F} v_{n} \vert ^{2}+ \vert \mathscr{F} u \vert ^{2}+2\mathscr{F} v_{n}\mathscr{F} u \bigr)\,d\xi \\ &\quad+ \int_{\mathbb{R}^{N}} \bigl(V(x)v_{n}^{2}+V(x)u^{2}+2V(x)v_{n}u \bigr)\,dx \\ &\quad- \int_{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p+1} \,dx+ \int_{\mathbb {R}^{N}}K(x) \vert u \vert ^{p+1}\,dx- \int_{\mathbb{R}^{N}}K(x) \vert u \vert ^{p+1}\,dx \\ &=\bigl\langle \varphi'(u),u\bigr\rangle + \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} v_{n} \vert ^{2}\,d\xi+ \int_{\mathbb{R}^{N}}V(x)v_{n}^{2}\, dx \\ &\quad- \int_{\mathbb{R}^{N}}K(x) \vert u_{n} \vert ^{p+1} \,dx+ \int_{\mathbb {R}^{N}}K(x) \vert u \vert ^{p+1} \,dx+o_{n}(1) \\ &\geq \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} v_{n} \vert ^{2}\,d\xi - \int_{\mathbb{R}^{N}}V^{-}(x)v_{n}^{2}\,dx \\ &\quad- \int_{\mathbb{R}^{N}}K(x) \bigl( \vert u_{n} \vert ^{p+1}- \vert u \vert ^{p+1} \bigr)\,dx+o_{n}(1). \end{aligned} $$

Thanks to (3.5) and Lemma 4.2 in [3], we have that

$$\lim_{n\rightarrow+\infty} \int_{\mathbb {R}^{N}}K(x)\bigl[ \vert u_{n} \vert ^{p+1}- \vert u \vert ^{p+1}\bigr]\,dx=\lim _{n\rightarrow+\infty } \int_{\mathbb{R}^{N}}K(x) \vert v_{n} \vert ^{p+1} \,dx. $$

So, we have that

$$ \begin{aligned}[b] \bigl\langle \varphi'(u_{n}),u_{n} \bigr\rangle & \geq \int _{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} v_{n} \vert ^{2}\,d\xi- \int_{\mathbb {R}^{N}}V^{-}(x)v_{n}^{2}\,dx \\ &\quad- \int_{\mathbb{R}^{N}}K(x) \vert v_{n} \vert ^{p+1} \,dx+o_{n}(1) \\ &= \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} v_{n} \vert ^{2}\,d\xi- \int _{\mathbb{R}^{N}}V^{-}(x)v_{n}^{2} \,dx \\ &\quad- \int_{\mathbb{R}^{N}} \bigl(K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert v_{n} \vert ^{p+1}\,dx \\ &\quad+ \int_{\mathbb{R}^{N}} \bigl(K^{-}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert v_{n} \vert ^{p+1}\,dx+o_{n}(1). \end{aligned} $$

(3.6)

Claim 1

$\int_{\mathbb{R}^{N}}V^{-}(x)v_{n}^{2}\,dx\rightarrow0$as$n\rightarrow+\infty$.

In fact, by $(V_{1})$, we have that $V^{-}(x)=0$ for all $|x|\geq R_{0}$. So, from $v_{n}\rightarrow0$ in $L^{q}_{\mathrm{loc}}(\mathbb {R}^{N})$, $2\leq q<2_{s}^{\ast}$, and $V\in L^{\infty}(\mathbb{R}^{N})$, we obtain $\int_{\mathbb{R}^{N}}V^{-}(x)v_{n}^{2}\,dx\rightarrow0$ as $n\rightarrow+\infty$.

Claim 2

$\int_{\mathbb{R}^{N}} (K^{+}(x)+\chi _{B(0,R_{1})}(x) )|v_{n}|^{p+1}\,dx\rightarrow0$as$n\rightarrow+\infty$.

In fact, thanks to $(K)$, we have that $K^{+}(x)=0$ for all $|x|> R_{1}$. So, by $K\in L^{\infty}(\mathbb{R}^{N})$ and $v_{n}\rightarrow 0$ in $L^{q}_{\mathrm{loc}}(\mathbb{R}^{N})$, $2\leq q<2_{s}^{\ast}$, we get

$$\int_{\mathbb{R}^{N}} \bigl(K^{+}(x)+\chi_{B(0,R_{1})}(x) \bigr) \vert v_{n} \vert ^{p+1}\,dx\rightarrow0 $$

as $n\rightarrow+\infty$.

From Claim 1, Claim 2, (3.3), and (3.6), we obtain that

$$0= \lim_{n\rightarrow+\infty} \bigl( \Vert v_{n} \Vert _{0}^{2}+\min(\beta ,1) \Vert v_{n} \Vert _{p+1}^{p+1} \bigr). $$

That is, $v_{n}\rightarrow0$ in E. The proof is complete. □

Lemma 3.3

Assume that$(V_{1})$–$(V_{2})$and$(K)$hold. Then, for each$k\in {\mathbb {N}}$, there exists$A_{k}\in\varGamma_{k}$such that

$$\sup_{u\in A_{k}}\varphi(u)< 0. $$

Proof

The proof is based on some ideas of Kajikiya [23] and is very similar to the one contained in [3]. For readers’ convenience, we give the proof. Let $R_{2}$ and $y_{0}$ be fixed as in $(K)$ and denote

$$D(R_{2})=\bigl\{ (x_{1},\ldots,x_{n})\in \mathbb{R}^{N}: \vert x_{i}-y_{i} \vert < R_{2}, 1\leq i\leq N\bigr\} . $$

Let $k\in {\mathbb {N}}$ be an arbitrary number and define $n=\min\{n\in {\mathbb {N}}:n^{N}\geq k\}$. By planes parallel to each face of $D(R_{2})$, let $D(R_{2})$ be equally divided into $n^{N}$ small parts $D_{i}$ with $1\leq i\leq n^{N}$. In fact, the length a of the edge $D_{i}$ is $\frac{R_{2}}{n}$. Let $F_{i}\subset D_{i}$ be new cubes such that $F_{i}$ has the same center as that of $D_{i}$. The faces of $F_{i}$ and $D_{i}$ are parallel, and the length of the edge of $F_{i}$ is $\frac{a}{2}$. Let $\phi_{i}$, $1\leq i \leq k$, satisfy: $\operatorname{supp}(\phi _{i})\subset D_{i}$; $\operatorname{supp}(\phi_{i})\cap \operatorname{supp}(\phi_{j})=\emptyset$ ($i\neq j$); $\phi_{i}(x)=1$ for $x\in F_{i}$; $0\leq\phi_{i}(x)\leq 1$, for all $x\in\mathbb{R}^{N}$. Let

$$\begin{aligned}& S^{k-1}=\Bigl\{ (t_{1}, \ldots,t_{k})\in {\mathbb {R}}^{k}: \max_{1\leq i\leq k} \vert t_{i} \vert =1\Bigr\} , \\& W_{k}=\Biggl\{ \sum_{i=1}^{k}t_{i} \phi_{i}(x):(t_{1},\ldots, t_{k})\in S^{k-1}\Biggr\} \subset E. \end{aligned}$$

(3.7)

According to the fact that the mapping $(t_{1},\ldots, t_{k})\rightarrow\sum_{i=1}^{k}t_{i}\phi_{i}$ from $S^{k-1}$ to $W_{k}$ is odd and homeomorphic, so $\gamma(W_{k})=\gamma(S^{k-1})=k$. Since $W_{k}$ is compact in E, then $\exists\alpha_{k}>0$ such that

$$\Vert u \Vert ^{2}\leq\alpha_{k},\quad \forall u\in W_{k}. $$

On the other hand, by Hölder’s inequality and Sobolev’s embedding, we have that

$$\Vert u \Vert _{2}\leq c \Vert u \Vert _{0}^{r} \Vert u \Vert _{p+1}^{1-r}\leq c \Vert u \Vert , $$

where $r=\frac{2_{s}^{\ast}(1-p)}{2(2_{s}^{\ast}-p-1)}$.

According to the above facts, there exists $c_{k}>0$ such that

$$\Vert u \Vert _{2}^{2}\leq c_{k}\quad \text{for all } u\in W_{k}. $$

Let $t>0$ and $u=\sum_{=1}^{k}t_{i}\phi_{i}(x)\in W_{k}$,

$$ \begin{aligned}[b] \varphi(tu)&=\frac{t^{2}}{2} \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi+\frac{t^{2}}{2} \int_{\mathbb {R}^{N}}V(x)u^{2}\,dx-\frac{1}{p+1}\sum _{i=1}^{k} \int _{D_{i}}K(x) \vert tt_{i}\phi_{i} \vert ^{p+1}\,dx\hspace{-12pt} \\ &\leq\frac{t^{2}}{2}\alpha_{k}+\frac{t^{2}}{2} \Vert V \Vert _{\infty }c_{k}-\frac{1}{p+1}\sum _{i=1}^{k} \int_{D_{i}}K(x) \vert tt_{i}\phi _{i} \vert ^{p+1}\,dx. \end{aligned} $$

(3.8)

From (3.7), there exists $j\in[1,k]$ such that $|t_{j}|=1$ and $|t_{i}|\leq1$ for $i\neq j$. So

$$ \begin{aligned}[b] \sum_{i=1}^{k} \int_{D_{i}}K(x) \vert tt_{i}\phi_{i} \vert ^{p+1}\, dx&= \int_{F_{j}}K(x) \vert tt_{j}\phi_{j} \vert ^{p+1}\,dx \\ &\quad+ \int_{D_{j}\setminus F_{j}}K(x) \bigl\vert tt_{j} \phi_{j}(x) \bigr\vert ^{p+1}\,dx+\sum _{i\neq j} \int_{D_{i}}K(x) \vert tt_{i}\phi_{i} \vert ^{p+1}\,dx.\hspace{-12pt} \end{aligned} $$

(3.9)

According to $\phi_{j}(x)=1$ for $x\in F_{j}$ and $|t_{j}|=1$, one has that

$$ \int_{F_{j}}K(x) \vert tt_{j}\phi_{j} \vert ^{p+1}\,dx= \vert t \vert ^{p+1} \int _{F_{j}}K(x)\,dx. $$

(3.10)

By $(K)$, one has that

$$ \int_{D_{j}\setminus F_{j}}K(x) \bigl\vert tt_{j} \phi_{j}(x) \bigr\vert ^{p+1}\,dx+\sum _{i\neq j} \int_{D_{i}}K(x) \vert tt_{i}\phi_{i} \vert ^{p+1}\,dx\geq0. $$

(3.11)

According to (3.8), (3.9), (3.10), and (3.11), we have that

$$\frac{\varphi(tu)}{t^{2}}\leq\frac{1}{2}\alpha_{k}+\frac{1}{2} \Vert V \Vert _{\infty}c_{k}-\frac{ \vert t \vert ^{p+1}}{(p+1)t^{2}}\inf _{1\leq i \leq k} \biggl( \int_{F_{i}}K(x)\,dx \biggr). $$

So,

$$\lim_{t\rightarrow0}\sup_{u\in W_{k}}\frac{\varphi (tu)}{t^{2}}=- \infty. $$

Hence, we can fix t small enough such that $\sup\{\varphi(u),u\in A_{k}\}<0$, where $A_{k}=tW_{k}\in\varGamma_{k}$. □

Lemma 3.4

Assume that$(V_{1})$–$(V_{2})$and$(K)$hold. Thenφis bounded from below.

Proof

By $(K)$, Hölder’s inequality and Sobolev’s embedding, as in the proof of Lemma 3.1, we have that

$$ \begin{aligned} \varphi(u)&=\frac{1}{2} \biggl( \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi+ \int_{\mathbb{R}^{N}}V(x)u^{2}\, dx \biggr)-\frac{1}{p+1} \int_{\mathbb{R}^{N}}K(x) \vert u \vert ^{p+1}\,dx \\ &\geq\frac{1}{2} \biggl( \int_{\mathbb{R}^{N}} \vert \xi \vert ^{2s} \vert \mathscr{F} u \vert ^{2}\,d\xi- \int_{\mathbb{R}^{N}}V^{-}(x)u^{2}\,dx \biggr)- \frac {1}{p+1} \int_{\mathbb{R}^{N}}K^{+}(x) \vert u \vert ^{p+1} \,dx \\ &\geq \biggl(\frac{1}{2}-\frac{S \Vert V^{-} \Vert _{\frac{N}{2s}}}{2} \biggr) \Vert u \Vert ^{2}_{0}-\frac{S^{\frac{p+1}{2}}}{p+1} \bigl\Vert K^{+} \bigr\Vert _{\frac {2_{s}^{\ast}}{2_{s}^{\ast}-p-1}} \Vert u \Vert ^{p+1}_{0}. \end{aligned} $$

Since $0< p<1$, we conclude the proof. □

Proof of Theorem 1.1

In fact, $\varphi(0)=0$ and φ is an even functional. Then by Lemmas 3.2, 3.3, and 3.4, conditions $(I_{1})$ and $(I_{2})$ of Theorem 2.1 are satisfied. Therefore, by Theorem 2.1, problem (1.1) possesses infinitely many nontrivial solutions converging to 0 with negative energy. □