# 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, 614, 1622, 24, 2729, 31, 3344, 4655] 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, 810, 12, 13, 16, 18, 2022, 24, 28, 33, 34, 3638, 44, 50, 5254], 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 . In , 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 , 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 . In fact, the PS condition was proved in  by concentration compactness principle. It is noticed that the PS condition plays important role in the proof of the main results in .

## 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 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 , 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 , 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

()

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 .

### Theorem 2.1

()

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:

1. (i)

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

2. (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 , 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  and is very similar to the one contained in . 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. □

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### Acknowledgements

The authors are thankful to the honorable reviewers and editors for their valuable reviewing of the manuscript.

### Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

## Funding

The paper is supported by the Natural Science Foundation of China (Grant nos. 11561043 and 11501318).

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### Contributions

All the authors have the same contribution. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Da-Bin Wang.

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