Skip to main content

Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

A Correction to this article was published on 08 May 2023

This article has been updated

Abstract

In this paper, we study the fractional Schrödinger equation

$$ \textstyle\begin{cases} (-\Delta )^{s}u+u=a(x) \vert u \vert ^{p-2}u+b(x) \vert u \vert ^{q-2}u, \\ u\in H^{s}(\mathbb{R}^{N}), \end{cases} $$

where \((-\Delta )^{s}\) denotes the fractional Laplacian of order \(s\in (0,1)\), \(N>2s\), \(2< p< q<2^{*}_{s}\), and \(2^{*}_{s}\) is the fractional critical Sobolev exponent. The weight potentials a or b is a sign-changing function and satisfies some valid assumptions. We obtain the existence of infinitely many solutions to the problem by the Nehari manifold.

1 Introduction and the main results

In this paper, we study the fractional Schrödinger equation

$$ \textstyle\begin{cases} (-\Delta )^{s}u+u=a(x) \vert u \vert ^{p-2}u+b(x) \vert u \vert ^{q-2}u, \\ u\in H^{s}(\mathbb{R}^{N}), \end{cases} $$
(1.1)

where the fractional Laplacian \((-\Delta )^{s}\) is defined by

$$ (-\Delta )^{s}\Psi (x)=C_{N,s}P.V. \int _{\mathbb{R}^{N}} \frac{\Psi (x)-\Psi (y)}{ \vert x-y \vert ^{N+2s}}\,dy,\quad \Psi \in \mathcal{S}\bigl( \mathbb{R}^{N}\bigr), $$

\(P.V\). stands for the Cauchy principal value, \(C_{N,s}\) is a normalizing constant, \(\mathcal{S}(\mathbb{R}^{N})\) is the Schwartz space of rapidly decaying functions, \(s\in (0,1)\), \(N>2s\), \(2< p< q<2^{*}_{s}\), and \(2^{*}_{s}\) is the fractional critical Sobolev exponent.

In the last few years, the time-dependent fractional Schrödinger equation has been studied extensively in the literature. It appears widely in optimization, finance, phase transitions, stratified materials, crystal dislocation, flame propagation, conservation laws, materials science, and water waves (see [5]). The standing waves have a wide range of applications in the real world. Musical instruments generally emit sound due to the standing wave generated by the vibration of a string. In the nonlinear fractional Schrödinger equation, the question of the existence and stability of the standing wave solution is an important research topic. It has been applied in many areas of physics, such as constructive quantum field theory, plasma physics, nonlinear optics, and so on. A basic motivation for the study of Eq. (1.1) arises in looking for the standing wave solutions of the type

$$ \Psi (x,t)=e^{-iEt/\varepsilon}u(x)$$

for the following time-dependent fractional Schrödinger equation:

$$ i\varepsilon \frac{\partial \Psi}{\partial t}=\varepsilon ^{2s}(- \Delta )^{s}\Psi +\bigl(V(x)+E\bigr)\Psi -f(x,\Psi ),\quad (x,t)\in{ \mathbb{R}^{N} \times \mathbb{R}}. $$
(1.2)

Equation (1.2), introduced by Laskin [16, 17], describes how the wave function of a physical system evolves over time. Unlike the classical Laplacian operator, the usual analysis tools for elliptic PDEs cannot be directly applied to (1.2) since \((-\Delta )^{s}\) is a nonlocal operator. Cafferelli and Silvestre [6] developed a powerful extension method, which transfers the nonlocal equation (1.2) into a local one on a half-space. Recently, Di Nezza, Palatucci, and Valdinoci [9] gave a survey on the fractional Sobolev spaces, which are more convenient for fractional Laplacian equations. Lee, Kim, Kim, and Scapellato [18] examined the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional p-Laplacian with concave–convex nonlinearities. Since then, there have been some works on the existence, multiplicity, and concentration phenomenon of solutions to the nonlinear fractional Schrödinger equation (1.2) and other differential problems driven by Laplace-type operators; see [1, 2, 7, 1013, 15, 1921, 25, 2832].

When \(s=1\), (1.1) is the classical semilinear elliptic equation in \(\mathbb{R}^{N}\) with sign-changing weight functions. Equations of this type have been studied extensively in recent years, mainly on bounded domains. Below we briefly describe some of this work. Berestycki et al. [3] studied the existence and nonexistence of positive solutions to the problem

$$ \textstyle\begin{cases} -\Delta u+m(x)u=a(x)u^{p},& x\in \Omega , \\ Bu(x)=0,& x\in \partial \Omega . \end{cases} $$
(1.3)

Here m and a may be sign-changing functions, \(1< p<2^{*}\), \(Bu=u\), and \(Bu=\partial _{\nu }u\) (respectively, the Dirichlet and Neumann boundary conditions). Brown and Zhang [4] considered a problem similar to (1.3) by splitting the Nehari manifold into three parts corresponding to local minima, local maxima, and points of inflection of the fibering map and then looked for minimizers of the energy functional on the first two parts. De Paiva [8] studied the problem

$$ \textstyle\begin{cases} -\Delta u=a(x)u^{p}+\lambda b(x)u^{q},& x\in \Omega , \\ u(x)=0,& x\in \partial \Omega , \end{cases} $$
(1.4)

where \(0< p<1<q\leq 2^{*}-1\), a changes sign, and \(b\geq 0\). He proved that there exists \(\lambda ^{*}\in (0,+\infty )\) such that (1.4) has at least one nonnegative solution for \(0<\lambda <\lambda ^{*}\) and there is no such solution for \(\lambda >\lambda ^{*}\). For an unbounded domain, we can mention Wu [27], who studied the multiplicity of positive solutions for the concave–convex elliptic equation

$$ \textstyle\begin{cases} -\Delta u+u=f_{\lambda}(x)u^{p-1}+g_{\mu}(x)u^{q-1},& x\in \mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$
(1.5)

where \(1< p<2<q<2^{*}\), the parameters \(\lambda ,\mu \geq 0\), \(f_{\lambda}=\lambda f_{+}+f_{-}\), \((f_{\pm}:=\max \{0,\pm f\})\) is sign-changing, and \(g_{\mu}(x)=a(x)+\mu b(x)\). Here the author used the idea of Brown and Zhang [4] mentioned above and has split the Nehari manifold into three parts considered separately. See also [14, 24], where related (singular) problems with sign-changing weights in \(\mathbb{R}^{N}\) are studied by similar techniques. To our best knowledge, there is no similar result for nonlocal problem (1.2) with sign-changing weight functions, so in this paper, we will fill this gap.

Our purpose here is to study the fractional Schrödinger equation (1.1). We are interested in the situation where one of the weight functions a and b is sign-changing, and unlike the papers mentioned above, we are mainly concerned with the existence of infinitely many solutions. In what follows, we assume that a and b satisfy some of the following hypotheses:

\((H_{1})\):

\(a\in L^{r}(\mathbb{R}^{N})\) and \(b\in L^{t}(\mathbb{R}^{N})\), where \(1<\frac{r}{r-1}<\frac{2^{*}_{s}}{p}\), \(1<\frac{t}{t-1}<\frac{2^{*}_{s}}{q}\);

\((H_{2})\):

\(a,b\in L^{\infty}(\mathbb{R}^{N})\), \(\limsup_{|x|\rightarrow \infty}a(x)\leq 0\), and \(\limsup_{|x|\rightarrow \infty}b(x)\leq 0\).

\((H_{3})\):

\(b\geq 0\) in \(\mathbb{R}^{N}\), and the set \(\{x\in \mathbb{R}^{N}: b(x)>0\}\) has a nonempty interior.

\((H_{4})\):

\(a\leq 0\) in \(\mathbb{R}^{N}\), and the set \(\{x\in \mathbb{R}^{N}: b(x)>0\}\) has a nonempty interior.

For example, \(a(x)=e^{-|x|}\sin |x|\) and \(b(x)=e^{-|x|}\) satisfy assumptions \((H_{1})\) or \((H_{2})\) and \((H_{3})\). \(a(x)=-e^{-|x|}\) and \(b(x)=e^{-|x|}\sin |x|\) satisfy assumptions \((H_{1})\) or \((H_{2})\) and \((H_{4})\).

Our main results of this paper is the following:

Theorem 1.1

Assume that \((H_{1})\) or \((H_{2})\) and \((H_{3})\) or \((H_{4})\) hold. Then problem (1.1) has infinitely many solutions.

Under the assumptions above, we prove that the Nehari manifold is closed and of class \(C^{2}\) and that the energy functional corresponding to problem (1.1) is bounded below. When \((H_{1})\) or \((H_{2})\) is satisfied, we show that the energy functional satisfies the Palais–Smale condition on the Nehari manifold, and then using some arguments based on the Krasnoselskii genus, we establish the existence of infinitely many solutions for problem (1.1).

This paper is organized as follows. In Sect. 2, we describe the functional setting to study problem (1.1) and prove some preliminary lemmas. In Sect. 3, we complete the proof of Theorem 1.1.

2 Variational settings and preliminary results

We denote by \(|\cdot |_{p}\) the usual norm of the space \(L^{p}(\mathbb{R}^{3})\), \(1\leq p<\infty \), by \(B_{r}(x)\) the open ball with center at x and radius r, and by C or \(C_{i}\) (\(i=1,2,\dots \)) positive constants that may change from line to line. By \(a_{n}\rightharpoonup a\) and \(a_{n}\rightarrow a\) we mean the weak and strong convergence, respectively, as \(n\rightarrow \infty \).

2.1 The functional space setting

Firstly, fractional Sobolev spaces are convenient for our problem, so we will give some sketches on them; a complete introduction can be found in [9]. We recall that for \(s\in (0,1)\), the fractional Sobolev space \(H^{s}(\mathbb{R}^{3})=W^{s,2}(\mathbb{R}^{3})\) is defined as follows:

$$ H^{s}\bigl(\mathbb{R}^{3}\bigr)=\biggl\{ u\in L^{2}\bigl(\mathbb{R}^{3}\bigr): \int _{\mathbb{R}^{3}}\bigl( \vert \xi \vert ^{2s} \bigl\vert {\mathcal{F}(u)} \bigr\vert ^{2}+ \bigl\vert { \mathcal{F}(u)} \bigr\vert ^{2}\bigr)\,d\xi < \infty \biggr\} $$

with the norm

$$ \Vert u \Vert ^{2}_{H^{s}(\mathbb{R}^{3})}= \int _{\mathbb{R}^{3}}\bigl( \vert \xi \vert ^{2s} \bigl\vert { \mathcal{F}(u)} \bigr\vert ^{2}+ \bigl\vert { \mathcal{F}(u)} \bigr\vert ^{2}\bigr)\,d\xi ,$$

where \(\mathcal{F}\) denotes the Fourier transform. We also define the homogeneous fractional Sobolev space \(\mathcal{D}^{s,2}(\mathbb{R}^{3})\) as the completion of \(\mathcal{C}_{0}^{\infty}(\mathbb{R}^{3})\) with respect to the norm

$$ \Vert u \Vert _{\mathcal{D}^{s,2}(\mathbb{R}^{3})}:= \biggl( \iint _{\mathbb{R}^{3} \times \mathbb{R}^{3}}\frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,dx \,dy \biggr)^{ \frac{1}{2}}=[u]_{H^{s}(\mathbb{R}^{3})}. $$

The fractional Laplacian \((-\Delta )^{s} u\) of a smooth function \(u:\mathbb{R}^{3}\rightarrow \mathbb{R}\) is defined by

$$ \mathcal{F}\bigl((-\Delta )^{s} u\bigr) (\xi )= \vert \xi \vert ^{2s}\mathcal{F}(u) (\xi ), \quad \xi \in \mathbb{R}^{3},$$

that is,

$$ \mathcal{F}(\phi ) (\xi )=\frac{1}{(2\pi )^{\frac{3}{2}}} \int _{ \mathbb{R}^{3}}e^{-i\xi \cdot x}\phi (x)\,dx$$

for functions ϕ in the Schwartz class. Also, \((-\Delta )^{s} u\) can be equivalently represented as (see [9])

$$ (-\Delta )^{s} u(x)=-\frac{1}{2}C(s) \int _{\mathbb{R}^{3}} \frac{u(x+y)+u(x-y)-2u(x)}{ \vert y \vert ^{3+2s}}\,dy, \quad x\in \mathbb{R}^{3},$$

where

$$ C(s)= \biggl( \int _{\mathbb{R}^{3}} \frac{(1-\cos\xi _{1})}{ \vert \xi \vert ^{3+2s}}\,d\xi \biggr)^{-1},\quad \xi =(\xi _{1}, \xi _{2},\xi _{3}).$$

Also, by the Plancherel formula in Fourier analysis we have

$$ [u]^{2}_{H^{s}(\mathbb{R}^{3})}=\frac{2}{C(s)} \bigl\Vert (-\Delta )^{ \frac{s}{2}}u \bigr\Vert ^{2}_{2}.$$

As a consequence, the norms on \(H^{s}(\mathbb{R}^{3})\)

$$\begin{aligned} &u\longmapsto \biggl( \int _{\mathbb{R}^{3}} \vert u \vert ^{2}\,dx+ \iint _{ \mathbb{R}^{3}\times \mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,dx \,dy \biggr)^{\frac{1}{2}}, \\ &u\longmapsto \biggl( \int _{\mathbb{R}^{3}}\bigl( \vert \xi \vert ^{2s} \bigl\vert {\mathcal{F}(u)} \bigr\vert ^{2}+ \bigl\vert { \mathcal{F}(u)} \bigr\vert ^{2}\bigr)\,d\xi \biggr)^{\frac{1}{2}}, \\ &u\longmapsto \biggl( \int _{\mathbb{R}^{3}} \vert u \vert ^{2}\,dx+ \bigl\Vert (- \Delta )^{ \frac{s}{2}}u \bigr\Vert ^{2}_{2} \biggr)^{\frac{1}{2}} \end{aligned}$$

are equivalent.

For the reader’s convenience, we consider the space \(X:=H^{s}(\mathbb{R}^{N})\) with the norm

$$ \Vert u \Vert = \biggl( \int _{\mathbb{R}^{3}} \vert u \vert ^{2}\,dx+ \bigl\Vert (- \Delta )^{\frac{s}{2}}u \bigr\Vert ^{2}_{2} \biggr)^{\frac{1}{2}}, $$

and we review the main embedding result for this class of fractional Sobolev spaces.

Lemma 2.1

([9])

Let \(0< s<1\). Then there exists a constant \(C=C(s)>0\) such that

$$ \Vert u \Vert ^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{3})}\leq C[u]^{2}_{H^{s}( \mathbb{R}^{3})}$$

for every \(u\in H^{s}(\mathbb{R}^{3})\), where \(2^{*}_{s}=\frac{6}{3-2s}\) is the fractional critical exponent. Moreover, the embedding \(X\hookrightarrow L^{r}(\mathbb{R}^{3})\) is continuous for any \(r\in [2,2^{*}_{s}]\) and is locally compact whenever \(r\in [2,2^{*}_{s})\).

Lemma 2.2

([22])

If \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb{R}^{3})\) and for some \(R>0\),

$$ \lim_{n\rightarrow \infty}\sup_{y\in \mathbb{R}^{3}} \int _{B_{R}(y)} \vert u_{n} \vert ^{2}\,dx=0, $$

then \(u_{n}\rightarrow 0\) in \(L^{r}(\mathbb{R}^{3})\) for all \(2< r<2_{s}^{*}\).

2.2 Properties of the Nehari manifold

Set \(X:=H^{s}(\mathbb{R}^{N})\), and let \(A, B: X\rightarrow \mathbb{R}\) be defined by

$$\begin{aligned}& A(u):= \int _{\mathbb{R}^{N}}a(x) \vert u \vert ^{p}\,dx, \end{aligned}$$
(2.1)
$$\begin{aligned}& B(u):= \int _{\mathbb{R}^{N}}b(x) \vert u \vert ^{q}\,dx. \end{aligned}$$
(2.2)

Remark 2.1

Since \(2< p< q<2^{*}_{s}\), it is easy to see that if \((H_{1})\) or \((H_{2})\) is satisfied, then \(A, B\in C^{2}(X,\mathbb{R})\).

It is clear that problem (1.1) is the Euler–Lagrange equation for the functional \(J:X \rightarrow \mathbb{R}\) defined by

$$ J(u)=\frac{1}{2} \Vert u \Vert ^{2}- \frac{1}{p} \int _{\mathbb{R}^{N}}a(x) \vert u \vert ^{p}\,dx- \frac{1}{q} \int _{\mathbb{R}^{N}}b(x) \vert u \vert ^{q}\,dx. $$
(2.3)

By this remark the action functional \(J\in C^{2}(X,\mathbb{R})\), and its critical points are weak solutions of problem (1.1). Moreover, for all \(u,v\in X\), we have

$$ \bigl\langle J^{\prime}(u),v\bigr\rangle = \int _{\mathbb{R}^{N}}(-\Delta )^{ \frac{s}{2}}u(-\Delta )^{\frac{s}{2}}v\,dx- \int _{\mathbb{R}^{N}}a(x) \vert u \vert ^{p-2}uvdx- \int _{\mathbb{R}^{3}}b(x) \vert u \vert ^{q-2}uvdx. $$

Hence in the following, we consider critical points of I using the variational method.

We first introduce the Nehari manifold associated with the functional J:

$$ \mathcal{N}:=\bigl\{ u\in X\backslash \{0\}:\bigl\langle J^{\prime}(u),u \bigr\rangle =0 \bigr\} =\bigl\{ u\in X\backslash \{0\}: \Vert u \Vert ^{2}=A(u)+B(u)\bigr\} . $$

Now we define the fibering map corresponding to \(u\in X\backslash \{0\}\) by setting \(\alpha _{u}(t)=J(tu)\), \(t>0\). Then

$$ \alpha _{u}(t)=\frac{t^{2}}{2} \Vert u \Vert ^{2}- \frac{t^{p}}{p}A(u)- \frac{t^{q}}{q}B(u), $$

and

$$ \alpha _{u}'(t)=t \Vert u \Vert ^{2}-t^{p-1}A(u)-t^{q-1}B(u). $$

Moreover, \(tu\in \mathcal{N}\) if and only if \(\alpha _{u}'(t)=0\).

Lemma 2.3

Suppose that either \((H_{1})\) or \((H_{2})\) is satisfied.

\((i)\):

If \((H_{3})\) holds, then the equation \(\alpha _{u}'(t)=0\) has only one solution \(t_{u}>0\), provided that \(A(u)>0\) or \(B(u)>0\). Moreover, \(J(u)>0\) for all \(u\in \mathcal{N}\).

\((\mathit{ii})\):

If \((H_{4})\) holds, then the equation \(\alpha _{u}'(t)=0\) has only one solution \(t_{u}>0\), provided that \(B(u)>0\). Moreover, \(J(u)>0\) for all \(u\in \mathcal{N}\).

Proof

Suppose \((H_{3})\) holds. If \(A(u)\leq 0\) and \(B(u)=0\), then \(\alpha _{u}'(t)>0\) for all \(t>0\). If \(B(u)>0\), or \(B(u)=0\) and \(A(u)>0\), then \(\alpha _{u}\) has a positive maximum.

Suppose \((H_{4})\) holds. Since now \(A(u)\leq 0\), \(\alpha _{u}'(t)>0\) for all \(t>0\) if \(B(u)\leq 0\), and \(\alpha _{u}\) has a positive maximum if \(B(u)>0\).

It remains to show that the equation \(\alpha _{u}'(t)=0\) has at most one solution. Since \(\alpha _{u}(t_{u})>0\), it will then follow that \(J(u)>0\) for all \(u\in \mathcal{N}\). Let \(\alpha _{u}'(t_{1})=\alpha _{u}'(t_{2})=0\) for \(t_{1},t_{2}>0\). We have

$$ \Vert u \Vert ^{2}=t^{p-2}_{1}A(u)+t_{2}^{q-2}B(u) $$

and

$$ \Vert u \Vert ^{2}=t^{p-2}_{2}A(u)+t_{2}^{q-2}B(u). $$

Therefore

$$ \bigl(t^{p-2}_{1}-t^{p-2}_{2} \bigr) \Vert u \Vert ^{2}=(t_{1}t_{2})^{p-2} \bigl(t_{1}^{q-p}-t_{2}^{q-p}\bigr)B(u). $$
(2.4)

From (2.4) we see that either \(t_{1}=t_{2}\) or \(B(u)<0\). However, in the second case, \(\alpha '_{t}\) is never 0 under our hypotheses. □

Lemma 2.4

Suppose that \((H_{1})\) and either \((H_{3})\) or \((H_{4})\) is satisfied. Then the Nehari manifold \(\mathcal{N}\) is a closed \(C^{2}\)-manifold. Moreover, \(\mathcal{N}\) is bounded away from zero.

Proof

Let \(u\in \mathcal{N}\). Then from the Sobolev and Hölder inequalities and \((H_{1})\) we have

$$ \Vert u \Vert ^{2}=A(u)+B(u)\leq \vert a \vert _{r} \vert u \vert ^{p}_{pr'}+ \vert b \vert _{s} \vert u \vert ^{q}_{qs'} \leq C_{1} \vert a \vert _{r} \Vert u \Vert ^{p}+C_{2} \vert b \vert _{s} \Vert u \Vert ^{q}, $$
(2.5)

where \(r'=\frac{r}{r-1}\), \(s'=\frac{s}{s-1}\), and \(C_{1}\), \(C_{2}\) are positive constants. Since \(2< p< q<2^{*}_{s}\), inequality (2.5) implies that the Nehari manifold is bounded away from 0.

Now we show that it is a closed \(C^{2}\)-manifold. Define \(\psi : X\rightarrow \mathbb{R}\) as

$$ \psi (u):=\bigl\langle J'(u),u \bigr\rangle = \Vert u \Vert ^{2}-A(u)-B(u). $$

By Remark (2.1), \(\psi \in C^{2}(X,\mathbb{R})M\) and by the definition of ψ, \(\mathcal{N}=\psi ^{-1}(0)\backslash \{0\}\). Since \(\mathcal{N}\) is bounded away from 0, \(\mathcal{N}\) is closed. If we show that every point of \(\mathcal{N}\) is regular for ψ, THEN the proof will be complete. Let \(u\in \mathcal{N}=\psi ^{-1}(0)\backslash \{0\}\). Then

$$ \Vert u \Vert ^{2}=A(u)+B(u) $$
(2.6)

and

$$ \bigl\langle \psi '(u),u \bigr\rangle =2 \Vert u \Vert ^{2}-pA(u)-qB(u). $$
(2.7)

It follows from (2.6) and (2.7) that

$$ \bigl\langle \psi '(u),u \bigr\rangle =(2-p) \Vert u \Vert ^{2}+(p-q)B(u). $$
(2.8)

Since, according to Lemma 2.3, \(B(u)\geq 0\) if \(u\in \mathcal{N}\), the right-hand side of (2.8) is negative. Hence every point of \(\mathcal{N}\) is regular for ψ. □

Lemma 2.5

Suppose that \((H_{2})\) and either \((H_{3})\) or \((H_{4})\) are satisfied. Then the Nehari manifold \(\mathcal{N}\) is a closed \(C^{2}\)-manifold. Moreover, \(\mathcal{N}\) is bounded away from zero.

Proof

Consider \(u\in \mathcal{N}\) and assume that \(a, b\in L^{\infty}(\mathbb{R}^{N})\). Then by the Sobolev inequality we have

$$ \Vert u \Vert ^{2}=A(u)+B(u)\leq \vert a \vert _{\infty} \vert u \vert _{p}^{p}+ \vert b \vert _{\infty} \vert u \vert _{q}^{q} \leq C_{1} \vert a \vert _{\infty} \Vert u \Vert ^{p}+C_{2} \vert b \vert _{\infty} \Vert u \Vert ^{q}, $$
(2.9)

where \(C_{1}\), \(C_{2}\) are positive constants. Using (2.9), we deduce that \(\mathcal{N}\) is bounded away from 0. The rest of the proof is the same as that of Lemma 2.4. □

A functional \(I\in C^{1}(X,\mathbb{R})\) is said to satisfy the Palais–Smale condition at the level \(c\in \mathbb{R}\) (the \((\mathit{PS})_{c}\)-condition for short) if every sequence \(\{u_{n}\}\subset X\) such that

$$ I(u_{n})\rightarrow c\quad \text{and}\quad I'(u_{n})\rightarrow 0 $$
(2.10)

admits a convergent subsequence. A sequence satisfying (2.10) is called a \((\mathit{PS})_{c}\)-sequence.

Lemma 2.6

Suppose that the assumptions of Lemma 2.4or 2.5are satisfied. Then \(u\neq 0\) is a critical point of J if and only if it is a critical point of \(J|_{\mathcal{N}}\), and \(\{u_{n}\}\subset \mathcal{N}\) is a \((\mathit{PS})_{c}\)-sequence for J if and only if it is a \((\mathit{PS})_{c}\)-sequence for \(J|_{\mathcal{N}}\).

Proof

It is clear that if \(u\neq 0\) is a critical point of J, then \(u\in \mathcal{N}\). Let \(u\in \mathcal{N}\). By (2.8) we know that \(\langle \psi '(u),u\rangle <0\), and therefore \(X=T_{u}\mathcal{N}\oplus \mathbb{R}u\). Since \(J'(u)|_{\mathbb{R}u}\equiv 0\) by the definition of \(\mathcal{N}\), the conclusion follows. □

3 Proof of Theorem 1.1

To prove Theorem 1.1, we need to recall the definition of the Krasnoselskii genus and an abstract multiplicity result in [23]. A set \(F\subset X\) is said to be symmetric if \(F=-F\). Let

$$ \Sigma :=\{ F\subset X: F \text{ is closed and symmetric}\}. $$

For \(F\neq \emptyset \) and \(F\in \Sigma \), the Krasnoselskii genus of F is the least integer n such that there exists an odd function \(f\in C(F,\mathbb{R}^{N}\backslash \{0\})\). The genus of F is denoted by \(\gamma (F)\). Set \(\gamma (\emptyset ):=0\) and \(\gamma (F):=\infty \) if for all n, there exists no f with the above property.

Theorem 3.1

([23])

Suppose \(J\in C^{1}(M,\mathbb{R})\) is an even functional on a complete symmetric \(C^{1,1}\)-manifold \(M\subset V\backslash \{0\}\) in some Banach space V. Suppose also that J satisfies the \((\mathit{PS})_{c}\)-condition for all \(c\in \mathbb{R}\) and is bounded from below on M. Let

$$ \hat{\gamma}(M):=\sup \bigl\{ \gamma (F): F\subset M \textit{ is compact and symmetric}\bigr\} . $$

Then the functional J possesses at least \(\hat{\gamma}(M)\leq \infty \) pairs of critical points.

We first need some auxiliary results.

Lemma 3.1

Suppose that assumption \((H_{1})\) holds. Then the functionals A and B defined by (2.1) and (2.2) are weakly continuous:

$$ A(u_{n})\rightarrow A(u)\quad \textit{and}\quad B(u_{n}) \rightarrow B(u)\quad \textit{as } u_{n}\rightharpoonup u. $$

Moreover, \(A', B': X\rightarrow X^{*}\) are completely continuous:

$$ A'(u_{n})\rightarrow A'(u)\quad \textit{and}\quad B'(u_{n})\rightarrow B'(u) \quad \textit{as } u_{n}\rightharpoonup u. $$

Proof

We only prove the lemma for A and \(A'\); for B, \(B'\), the proof is similar. Let \(u_{n}\in X\) and \(u_{n}\rightharpoonup u\). Using the Rellich–Kondrachov theorem, up to a subsequence, we have

$$ \begin{aligned} &u_{n}\rightharpoonup u\quad \text{in } X, \\ &u_{n}\rightarrow u\quad \text{in } L^{l}_{\mathrm{loc}} \bigl(\mathbb{R}^{N}\bigr),2\leq l< 2_{s}^{*}, \\ &u_{n}(x)\rightarrow u(x)\quad \text{a.e. in } \mathbb{R}^{N}. \end{aligned} $$
(3.1)

Let \(w_{n}:=|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\). By (3.1) we get

$$ w_{n}(x)\rightarrow 0\quad \text{a.e. in } \mathbb{R}^{N}. $$
(3.2)

Assumption \((H_{1})\) and the Sobolev inequality imply

$$ \vert w_{n} \vert ^{\frac{p}{p-1}}\leq C_{1}\bigl( \vert u_{n} \vert ^{p}+ \vert u \vert ^{p}\bigr)\in L^{ \frac{r}{r-1}}\bigl(\mathbb{R}^{N} \bigr). $$
(3.3)

The boundedness of \(\{u_{n}\}\) in X, (3.2), and (3.3) imply that \(\{|w_{n}|^{\frac{p}{p-1}}\}\) is bounded in \(L^{\frac{r}{r-1}}(\mathbb{R}^{N})\) and up to a subsequence,

$$ \vert w_{n} \vert ^{\frac{p}{p-1}}\rightharpoonup 0 \quad \text{in } L^{ \frac{r}{r-1}}\bigl(\mathbb{R}^{N}\bigr). $$
(3.4)

Now let \(v\in X\) with \(\|v\|\leq 1\). Using the Hölder and the Sobolev inequalities, we deduce that

$$ \begin{aligned} \bigl\vert \bigl\langle A'(u_{n})-A'(u), v\bigr\rangle \bigr\vert &= \biggl\vert \int _{\mathbb{R}^{N}}a(x)w_{n}\,dx \biggr\vert \\ &\leq \biggl( \int _{\mathbb{R}^{N}} \bigl\vert a(x) \bigr\vert \vert v \vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggl( \int _{\mathbb{R}^{N}} \bigl\vert a(x) \bigr\vert \vert w_{n} \vert ^{\frac{p}{p-1}} \biggr)^{ \frac{p-1}{p}} \\ &\leq \vert a \vert _{r}^{\frac{1}{p}} \vert v \vert _{\frac{pr}{r-1}} \biggl( \int _{ \mathbb{R}^{N}} \bigl\vert a(x) \bigr\vert \vert w_{n} \vert ^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \\ &\leq C \vert a \vert _{r}^{\frac{1}{p}} \Vert v \Vert \biggl( \int _{\mathbb{R}^{N}} \bigl\vert a(x) \bigr\vert \vert w_{n} \vert ^{ \frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \\ &\leq C \vert a \vert _{r}^{\frac{1}{p}} \biggl( \int _{\mathbb{R}^{N}} \bigl\vert a(x) \bigr\vert \vert w_{n} \vert ^{ \frac{p}{p-1}} \biggr)^{\frac{p-1}{p}}, \end{aligned} $$

where \(C>0\) is a constant. Since \(|w_{n}|^{\frac{p}{p-1}}\rightharpoonup 0\) in \(L^{\frac{r}{r-1}}(\mathbb{R}^{N})\) and a is in the dual space of \(L^{\frac{r}{r-1}}(\mathbb{R}^{N})\), the right-hand side above goes to 0 uniformly with respect to \(\|v\|\leq 1\), which this implies that \(A'\) is completely continuous. By the definition of A,

$$ A(u)=\frac{1}{p}\bigl\langle A'(u), u\bigr\rangle . $$

Thus

$$ A(u)-A(u_{n})=\frac{1}{p}\bigl\langle A'(u), u \bigr\rangle -\frac{1}{p}\bigl\langle A'(u_{n}), u_{n}\bigr\rangle \rightarrow 0 $$

by the complete continuity of \(A'\). This proves the weak continuity of A. □

Let

$$ a^{+}(x):=\max \bigl\{ 0,a(x)\bigr\} ,\qquad a^{-}(x):=\max \bigl\{ 0,-a(x)\bigr\} , $$
(3.5)

and define \(b^{\pm}(x)\) similarly. Also, put

$$ A_{\pm}(u):= \int _{\mathbb{R}^{N}}a^{\pm}(x) \vert u \vert ^{p}\,dx,\qquad B_{\pm}(u):= \int _{\mathbb{R}^{N}}b^{\pm}(x) \vert u \vert ^{q}\,dx. $$

Lemma 3.2

Suppose that assumption \((H_{2})\) holds. Then \(A_{+}\), \(B_{+}\) are weakly continuous, and \(A'_{+}, B'_{+}: X\rightarrow X^{*}\) are completely continuous.

Proof

We only prove the lemma for \(A_{+}\), \(A'_{+}\); for \(B_{+}\), \(B'_{+}\), the proof is similar. Let \(u_{n}\in X\) and \(u_{n}\rightharpoonup u\). Using the Rellich–Kondrachov theorem, up to a subsequence, we have

$$ \begin{aligned} &u_{n}\rightharpoonup u\quad \text{in } X, \\ &u_{n}\rightarrow u\quad \text{in } L^{l}_{\mathrm{loc}} \bigl(\mathbb{R}^{N}\bigr),2\leq l< 2_{s}^{*}, \\ &u_{n}(x)\rightarrow u(x)\quad \text{a.e. in } \mathbb{R}^{N}. \end{aligned} $$
(3.6)

As in the preceding proof, let \(w_{n}:=|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\). We see from the Krasnoselskii theorem (see [26]) and (3.6) that

$$ w_{n}\rightarrow 0\quad \text{in } L^{\frac{p}{p-1}}_{\mathrm{loc}}\bigl(\mathbb{R}^{N}\bigr). $$
(3.7)

It follows from \((H_{2})\) that for any \(\varepsilon >0\), there exists \(R>0\) such that

$$ a^{+}(x)< \varepsilon \quad \text{whenever } \vert x \vert \geq R. $$
(3.8)

Using the Hölder and the Sobolev inequalities and (3.7), we obtain

$$ \begin{aligned} \sup_{ \Vert v \Vert \leq 1} \biggl\vert \int _{ \vert x \vert \leq R}a^{+}(x)w_{n}v\,dx \biggr\vert & \leq \bigl\vert a^{+} \bigr\vert _{\infty} \biggl( \int _{ \vert x \vert \leq R} \vert w_{n} \vert ^{\frac{p}{p-1}}\,dx \biggr)^{\frac{p-1}{p}} \biggl( \int _{ \vert x \vert \leq R} \vert v \vert ^{p}\,dx \biggr)^{ \frac{1}{p}} \\ &\leq C_{1} \biggl( \int _{ \vert x \vert \leq R} \vert w_{n} \vert ^{\frac{p}{p-1}}\,dx \biggr)^{ \frac{p-1}{p}} \\ &\rightarrow 0 \end{aligned} $$
(3.9)

as \(n\rightarrow \infty \). By inequality (3.3), \(\{w_{n}\}\) is bounded in \(L^{\frac{p}{p-1}}(\mathbb{R}^{N})\), hence inequality (3.8), the Hölder and the Sobolev inequalities imply that there exists a constant \(C_{2}>0\), independent of \(\varepsilon >0\), such that

$$ \sup_{ \Vert v \Vert \leq 1} \biggl\vert \int _{ \vert x \vert >R}a^{+}(x)w_{n}v\,dx \biggr\vert \leq C_{2} \varepsilon . $$
(3.10)

Using (3.9) and (3.10), we deduce that

$$ \sup_{ \Vert v \Vert \leq 1} \bigl\vert \bigl\langle A'_{+}(u_{n})-A'_{+}(u),v \bigr\rangle \bigr\vert =\sup_{ \Vert v \Vert \leq 1} \biggl\vert \int _{\mathbb{R}^{N}}a^{+}(x)w_{n}v\,dx \biggr\vert \rightarrow 0. $$

Since

$$ A_{+}(u)=\frac{1}{p}\bigl\langle A'_{+}(u),u \bigr\rangle , $$

we see as in the proof of Lemma 3.2 that \(A_{+}\) is weakly continuous. □

Lemma 3.3

Suppose that \((H_{1})\) or \((H_{2})\) and \((H_{3})\) or \((H_{4})\) are satisfied. Then the functional J satisfies the \((\mathit{PS})_{c}\)-condition on \(\mathcal{N}\) for all \(c\in \mathbb{R}\).

Proof

Let \(c\in \mathbb{R}\), and let \(\{w_{n}\}\subset \mathcal{N}\) be a \((\mathit{PS})_{c}\)-sequence. Then

$$ \Vert u_{n} \Vert ^{2}=A(u_{n})+B(u_{n}) $$
(3.11)

and

$$ J'(u_{n})\rightarrow 0,\qquad J(u_{n}) \rightarrow c. $$

If \((H_{3})\) holds, then \(B(u)\geq 0\), and we have, using (3.11) and the boundedness of \(J(u_{n})\),

$$ \begin{aligned} c+1&\geq J(u) \\ &=\frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{p}A(u)- \frac{1}{q}B(u) \\ &=\biggl(\frac{1}{2}-\frac{1}{p}\biggr) \Vert u \Vert ^{2}+\biggl(\frac{1}{p}-\frac{1}{q}\biggr)B(u) \\ &\geq \biggl(\frac{1}{2}-\frac{1}{p}\biggr) \Vert u \Vert ^{2} \end{aligned} $$
(3.12)

for all n large enough.

If \((H_{4})\) holds, then \(A(u_{n})\leq 0\), and using (3.11) again, we have

$$ \begin{aligned} c+1&\geq J(u) \\ &=\frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{p}A(u)- \frac{1}{q}B(u) \\ &=\biggl(\frac{1}{2}-\frac{1}{q}\biggr) \Vert u \Vert ^{2}-\biggl(\frac{1}{p}-\frac{1}{q}\biggr)A(u) \\ &\geq \biggl(\frac{1}{2}-\frac{1}{q}\biggr) \Vert u \Vert ^{2} \end{aligned} $$
(3.13)

for all n large enough. Hence in both cases, \(\{u_{n}\}\) is a bounded sequence. So there exists \(u\in X\) such that passing to a subsequence, \(u_{n}\rightharpoonup u\). Since \(J(u_{n})\rightarrow 0\), it is easy to see that \(J(u)=0\), and it follows that

$$ \bigl\langle J'(u_{n})-J'(u), u_{n}-u\bigr\rangle \rightarrow 0 $$

or, equivalently,

$$ \Vert u_{n}-u \Vert ^{2}-\bigl\langle A'(u_{n})-A'(u), u_{n}-u\bigr\rangle -\bigl\langle B'(u_{n})-B'(u), u_{n}-u\bigr\rangle \rightarrow 0. $$
(3.14)

If \((H_{1})\) holds, then by Lemma 3.1, \(A'(u_{n})\rightarrow A'(u)\) and \(B'(u_{n})\rightarrow B'(u)\). Thus from (3.14) we get \(u_{n}\rightarrow u\) in X.

If \((H_{2})\) holds, then since the function \(u\mapsto |u|^{t}\) is convex for \(t\geq 2\) (in particular, for \(t=p\) and q), we have \((|v|^{t-2}v-|u|^{t-2}u)(v-u)\geq 0\). Therefore, using (3.14), we have

$$ \begin{aligned} & \Vert u_{n}-u \Vert ^{2}- \bigl\langle A'_{+}(u_{n})-A'_{+}(u), u_{n}-u \bigr\rangle -\bigl\langle B'_{+}(u_{n})-B'_{+}(u), u_{n}-u\bigr\rangle \\ &\quad\leq \Vert u_{n}-u \Vert ^{2}-\bigl\langle A'(u_{n})-A'(u), u_{n}-u\bigr\rangle - \bigl\langle B'(u_{n})-B'(u), u_{n}-u\bigr\rangle \\ &\quad\rightarrow 0. \end{aligned} $$

Since \(A'_{+}(u_{n})\rightarrow A'_{+}(u)\) and \(B'_{+}(u_{n})\rightarrow B'_{+}(u)\), we see that \(u_{n}\rightarrow u\) in X again. □

Proof of Theorem 1.1

By Lemmas 2.32.5 and 3.3, \(\mathcal{N}\) is a closed symmetric \(C^{2}\)-manifold, \(J(u)>0\) for all \(u\in \mathcal{N}\), and J satisfies the \((\mathit{PS})_{c}\)-condition on \(\mathcal{N}\) for all \(c\in \mathbb{R}\). If we show that for any \(j\geq 1\), there exists a symmetric compact set \(F_{j}\subset \mathcal{N}\) such that \(\gamma (F_{j})\geq j\), then the conclusion will follow from Lemma 2.6 and Theorem 3.1.

Let \(j\geq 1\), let \(X_{j}\) be a subspace spanned by j linearly independent functions \(v_{k}\in C^{\infty}_{0}(\mathbb{R}^{N})\) with \(\operatorname{supp} v_{k}\subset \{x\in \mathbb{R}^{N}: b(x)>0\}\), and let

$$ S^{j-1}:=X_{j}\cap \bigl\{ x\in X: \Vert u \Vert =1\bigr\} . $$

Then we have \(B(u)>0\) for all \(u\in S^{j-1}\), and it follows from Lemma 2.3 that the equation \(\alpha '_{u}(t)=0\) has exactly one solution \(t_{u}\in (0,\infty )\). Hence the mapping \(\varphi : S^{j-1}\rightarrow \mathcal{N}\) given by \(\varphi (u):=t_{u}u\) is well defined, and it is obviously odd. If it is continuous, then \(F_{j}:=\varphi (S^{j-1})\) is homeomorphic to \(S^{j-1}\), and it follows from the properties of genus that \(\gamma (F_{j})=\gamma (S^{j-1})=j\).

We need to show that \(u\mapsto t_{u}\) is continuous. An easy computation shows that if the (necessary and sufficient) conditions for the existence of \(t_{u}\) given in Lemma 2.3 are satisfied, then \(\alpha _{u}''(t)<0\) for \(t=t_{u}\). Hence the continuity of \(t_{u}\) follows from the implicit function theorem. □

Availability of data and materials

Not applicable.

Change history

References

  1. Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in \(\mathbb{R}^{N}\) via penalization method. Calc. Var. Partial Differ. Equ. 55(3), 19 (2016)

    Article  MATH  Google Scholar 

  2. Ambrosio, V.: Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method. Ann. Mat. Pura Appl. (4) 196(6), 2043–2062 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2(4), 553–572 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193(2), 481–499 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Berlin (2016)

    MATH  Google Scholar 

  6. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, X., Wang, Z.: Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256(8), 2965–2992 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Paiva, F.O.: Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity. J. Funct. Anal. 261(9), 2569–2586 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. A 142(6), 1237–1262 (2012)

    Article  MATH  Google Scholar 

  11. Gu, G., Yu, Y., Zhao, F.: The least energy sign-changing solution for a nonlocal problem. J. Math. Phys. 58(5), 051505, 11 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gu, G., Zhang, W., Zhao, F.: Infinitely many positive solutions for a nonlocal problem. Appl. Math. Lett. 84, 49–55 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gu, G., Zhang, W., Zhao, F.: Infinitely many sign-changing solutions for a nonlocal problem. Ann. Mat. Pura Appl. (4) 197(5), 1429–1444 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jalilian, Y., Szulkin, A.: Infinitely many solutions for semilinear elliptic problems with sign-changing weight functions. Appl. Anal. 93(4), 756–770 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jin, T., Yang, Z.: The fractional Schrödinger–Poisson systems with infinitely many solutions. J. Korean Math. Soc. 57(2), 489–506 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66(5), 056108 (2002)

    Article  MathSciNet  Google Scholar 

  18. Lee, J., Kim, J.-M., Kim, Y.-H., Scapellato, A.: On multiple solutions to a nonlocal fractional \(p(\cdot )\)-Laplacian problem with concave–convex nonlinearities. Adv. Cont. Discr. Mod. 14, 25 (2022)

    MathSciNet  Google Scholar 

  19. Molica Bisci, G., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, vol. 162. Cambridge University Press, Cambridge (2016)

    Book  MATH  Google Scholar 

  20. Papageorgiou, N.S., Scapellato, A.: Nonlinear singular problems with convection. J. Differ. Equ. 296, 493–511 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Papageorgiou, N.S., Scapellato, A.: Positive solutions for anisotropic singular Dirichlet problems. Bull. Malays. Math. Sci. Soc. 45(3), 1141–1168 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^{N}\). J. Math. Phys. 54(3), 031501, 17 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Struwe, M.: Variational Methods. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  24. Szulkin, A., Waliullah, S.: Infinitely many solutions for some singular elliptic problems. Discrete Contin. Dyn. Syst. 33(1), 321–333 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Teng, K.: Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent. J. Differ. Equ. 261(6), 3061–3106 (2016)

    Article  MATH  Google Scholar 

  26. Willem, M.: Minimax Theorems. Birkhäuser Boston, Inc., Boston (1996)

    Book  MATH  Google Scholar 

  27. Wu, T.: Multiple positive solutions for a class of concave-convex elliptic problems in \(\mathbb{R}^{N}\) involving sign-changing weight. J. Funct. Anal. 258(1), 99–131 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang, Z., Yu, Y., Zhao, F.: The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system. Math. Nachr. 292(8), 1837–1868 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, Z., Yu, Y., Zhao, F.: Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent. Commun. Contemp. Math. 21(6), 1850027, 46 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, Z., Zhang, W., Zhao, F.: Existence and concentration results for fractional Schrödinger–Poisson system via penalization method. Electron. J. Differ. Equ. 14, 31 (2021)

    MATH  Google Scholar 

  31. Yang, Z., Zhao, F.: Three solutions for a fractional Schrödinger equation with vanishing potentials. Appl. Math. Lett. 76, 90–95 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, Z., Zhao, F.: Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Adv. Nonlinear Anal. 10(1), 732–774 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referee for his/her careful reading of our manuscript and the useful comments made for its improvement.

Funding

Y. Chen was supported by National Science Foundation of China (12161007), Guangxi science and technology base and talent project (AD21238019), and Doctoral Foundation of Guangxi University of Science and Technology (20Z30). B. Jin was supported by Basic ability improvement project of young and middle-aged teachers in Guangxi Universities (2017KY1384, 2021KY1721).

Author information

Authors and Affiliations

Authors

Contributions

B. Jin wrote the main manuscript text, and Y. Chen read and approved the final manuscript.

Corresponding author

Correspondence to Baoxia Jin.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Jin, B. Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions. Bound Value Probl 2022, 86 (2022). https://doi.org/10.1186/s13661-022-01667-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-022-01667-1

MSC

Keywords