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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.

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 [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].

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 [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:

  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.

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. □

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Acknowledgements

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

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

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Guan, W., Wang, D. & Hao, X. Infinitely many solutions for a class of sublinear fractional Schrödinger equations with indefinite potentials. J Inequal Appl 2020, 61 (2020). https://doi.org/10.1186/s13660-020-02326-8

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MSC

  • 35J20
  • 35J60
  • 47J30

Keywords

  • Fractional Schrödinger equation
  • Indefinite potential
  • Symmetric mountain pass theorem