Skip to main content

Multiplicity of solutions for fractional Schrödinger equations with perturbation

Abstract

In this paper, we investigate a class of fractional Schrödinger equations with perturbation. By using the mountain pass theorem and Ekeland’s variational principle, we see that such equations possess two solutions. Recent results in the literature are generalized and significantly improved.

1 Introduction

In this paper, we consider the following class of fractional Schrödinger equations:

$$ (-\Delta)^{\alpha}u+V(x)u=f(x,u)+\lambda h(x)|u|^{p-2}u,\quad x\in \mathbb{R}^{N}, $$
(1.1)

where \(0<\alpha<1\), \(2\alpha< N\), \(1\leq p<2\), \(f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R})\), \(h\in L^{\frac{2}{2-p}}(\mathbb{R}^{N})\), \(V\in C(\mathbb{R}^{N},\mathbb{R})\), and \((-\Delta)^{\alpha}u\) is defined pointwise for x in \(\mathbb{R}^{N}\) by

$$(-\Delta)^{\alpha}u(x)=-\frac{1}{2}\int_{\mathbb{R}^{N}} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2\alpha}}\,dy $$

along any rapidly decaying function u of class \(C^{\infty}(\mathbb {R}^{N})\); see Lemma 3.5 of [1].

Recently, a lot of attention has been focused on the study of fractional and non-local problems; see [26]. This may be due to its concrete applications in different fields, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, deblurring and denoising of images, and so on; see [1, 710]. For standing wave solutions of fractional Schrödinger equations in the whole space \(\mathbb{R}^{N}\), there were also many works; see [1120]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. The fractional quantum mechanics has been discovered as a result of expanding the Feynman path integral, from Brownian-like to Lévy-like quantum mechanical paths. In [11], Laskin formulated the fractional Schrödinger equations as follows:

$$ i\partial_{t}\psi=(-\Delta)^{\alpha}\psi+V(x)\psi-| \psi|^{p-1}\psi,\quad x\in\mathbb{R}^{N}, t\in \mathbb{R}, $$
(1.2)

where \(0<\alpha<1\), ψ is the wavefunction and \(V(x)\) denotes the potential energy. We let \(\psi(x,t)=e^{i\omega t}u(x)\) be standing waves solutions for (1.2). Then u is a solution of an equation of type of (1.1).

For the fractional Schrödinger equations, variational methods are available. In [14], Felmer et al. studied the existence and regularity of solutions for a class of fractional Schrödinger equations under the Ambrosetti-Rabinowitz condition, i.e., there exists \(\theta>2\) such that

$$0< \theta F(x,t)\leq tf(x,t). $$

In [15], Secchi obtained the existence of ground state solutions of a class of fractional Schrödinger equations under the Ambrosetti-Rabinowitz condition and the following condition:

(V0):

\(V\in C(\mathbb{R}^{N})\), \(\inf_{x\in\mathbb{R}^{N}}V(x)=V_{0}>0\) and \(\lim_{|x|\rightarrow\infty }V(x)=\infty\).

In [19], Torres studied the existence of solutions for the following equations:

$$ (-\Delta)^{\alpha}u+V(x)u=f(u)+ h(x),\quad x\in\mathbb{R}^{N}, $$
(1.3)

under the conditions of (V0) and the Ambrosetti-Rabinowitz condition for f.

As far as we know, there are few works on problem (1.1), of which nonlinearity involves a combination of superlinear or asymptotically linear terms and a sublinear perturbation. Motivated by the above facts, we investigate this case in this paper.

Before stating our results we introduce some notations. Throughout this paper, we denote by \(\| \|_{r}\) the \(L^{r}\)-norm, \(2\leq r\leq\infty \), and \(h^{\pm}=\max\{\pm h,0\}\). If we take a subsequence of a sequence \(\{u_{n}\}\) we shall denote it again by \(\{u_{n}\}\).

Now we state our main result.

Theorem 1.1

Assume that \(h\in L^{\frac{2}{2-p}}\setminus\{ 0\}\) with \(h^{+}\neq0\), (V0), and the following conditions are satisfied:

(F1) \(f(x,s)\) is a continuous function on \(\mathbb {R}^{N}\times\mathbb{R}\) such that \(f(x,s)\equiv0\) for all \(s<0\) and \(x\in\mathbb{R}^{N}\). Moreover, there exists \(b\in L^{\infty}(\mathbb {R}^{N},\mathbb{R}^{+})\) with \(|b|_{\infty}<\frac{S_{2}^{2}}{2}\) such that

$$\lim_{s\rightarrow0^{+}}\frac{f(x,s)}{s^{k}}=b(x) \quad\textit{uniformly in } x \in\mathbb{R}^{N}, $$

and

$$\frac{f(x,s)}{s^{k}}\geq b(x) \quad\forall s>0 \textit{ and } x\in \mathbb{R}^{N}, $$

where \(S_{r}\) is the best constant for the embedding of X in \(L^{r}(\mathbb{R}^{N})\); see Lemma  2.2 and Remark  2.1 in Section  2;

(F2) there exists \(q\in L^{\infty}(\mathbb{R}^{N},\mathbb {R}^{+})\) with \(|q|_{\infty}>c_{0}\) such that

$$\lim_{s\rightarrow\infty}\frac{f(x,s)}{s^{k}}=q(x) \quad\textit{uniformly in } x \in\mathbb{R}^{N}, $$

where \(c_{0}\) is defined by (2.1) in Section  2;

(F3) there exist two constants θ, \(d_{0}\) satisfying \(\theta>2\) and \(0\leq d_{0}<\frac{S_{2}^{2}(\theta-2)}{2\theta}\) such that

$$F(x,s)-\frac{1}{\theta}f(x,s)s\leq d_{0}s^{2} \quad \forall s>0 \textit{ and } x\in\mathbb{R}^{N}, $$

where \(F(x,s)=\int_{0}^{s}f(x,\tau)\,d\tau\).

Then we have the following results:

  1. (i)

    if \(k=1\) and \(\mu<1\) with

    $$\begin{aligned} \mu={}&\inf \biggl\{ \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac {|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x)u(x)^{2}\,dx\Big| u\in H^{\alpha} \bigl(\mathbb{R}^{N} \bigr),\\ &{}\int_{\mathbb {R}^{N}}q(x)u(x)^{2}\,dx=1 \biggr\} , \end{aligned}$$

    then there exists \(\Lambda>0\) such that for every \(\lambda\in(0,\Lambda )\), problem (1.1) has at least two nontrivial solutions;

  2. (ii)

    if \(1< k<2_{\alpha}^{\ast}-1\), then there exists \(\Lambda >0\) such that for every \(\lambda\in(0,\Lambda)\), problem (1.1) has at least two nontrivial solutions, where \(2_{\alpha}^{\ast}=\frac {2N}{N-2\alpha}\).

Remark 1.1

Theorem 1.1 extends the perturbation h in [19] to the case \(\lambda h(x)|u|^{p-2}u\) and (F3) is weaker than the Ambrosetti-Rabinowitz condition. Moreover, our f is allowed to be asymptotically linear at infinity when \(k=1\), which is not the same as that in [20], where they need \(\limsup_{s\rightarrow0^{+}}\frac {f(x,s)}{s}<\liminf_{s\rightarrow+\infty}\frac{f(x,s)}{s}\).

The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we give the proof of our main results.

2 Preliminaries

In order to prove our main results, we first give some properties of space X on which the variational setting for problem (1.1) is defined. Let

$$H=H^{\alpha} \bigl(\mathbb{R}^{N} \bigr):= \biggl\{ u\in L^{2} \bigl(\mathbb{R}^{N} \bigr):\int_{\mathbb {R}^{N}} \int_{\mathbb{R}^{N}}\frac{|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha }}\,dx\,dz< \infty \biggr\} $$

with the inner product and the norm

$$\langle u,v\rangle_{H}=\int_{\mathbb{R}^{N}}\int _{\mathbb{R}^{N}}\frac {[u(x)-u(z)][v(x)-v(z)]}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}u(x)v(x)\,dx,\quad \|u\|_{H}=\langle u,u\rangle_{H}^{\frac{1}{2}}. $$

Letting

$$X= \biggl\{ u\in H^{\alpha} \bigl(\mathbb{R}^{N} \bigr):\int _{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz +\int _{\mathbb{R}^{N}}V(x)u^{2}(x)\,dx< +\infty \biggr\} , $$

then X is a Hilbert space with the inner product

$$\langle u,v\rangle=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac {[u(x)-u(z)][v(x)-v(z)]}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x)u(x)v(x)\,dx $$

and the corresponding norm \(\|u\|^{2}=\langle u,u\rangle\). Note that

$$X\subset H^{\alpha} \bigl(\mathbb{R}^{N} \bigr) $$

and

$$X\subset L^{r} \bigl(\mathbb{R}^{N} \bigr) $$

for all \(r\in[2,2_{\alpha}^{\ast}]\) with the embedding being continuous. It is easy to get the following lemma.

Lemma 2.1

Assume that the condition (V0) holds. Then there exists a constant \(c_{0}>0\) such that

$$ \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac {|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+ \int_{\mathbb {R}^{N}}V(x)u^{2}\,dx\geq c_{0}\|u \|^{2}_{H},\quad \forall u\in H^{\alpha } \bigl( \mathbb{R}^{N} \bigr). $$
(2.1)

Lemma 2.2

(see [15, 19])

Assume that the condition (V0) holds. Then X is compactly embedded in \(L^{r}(\mathbb{R}^{N})\) for all \(r\in[2,2_{\alpha}^{\ast})\).

Remark 2.1

By Lemma 2.2, we have

$$S_{r}\|u\|_{r}\leq\|u\|, $$

where \(S_{r}\) is the best constants for the embedding of X in \(L^{r}(\mathbb{R}^{N})\).

Now we begin describing the variational formulation of problem (1.1). Consider the functional \(J:X\rightarrow\mathbb{R}\) defined by

$$ J(u)=\frac{1}{2}\|u\|^{2}-\int_{\mathbb{R}^{N}}F(x,u)\,dx- \frac{\lambda }{p}\int_{\mathbb{R}^{N}}h(x)|u|^{p}\,dx. $$
(2.2)

By the continuity of f, g and Lemma 2.2, \(J\in C^{1}(X,\mathbb{R})\) and its derivative is given by

$$\begin{aligned} J'(u)v={}&\int_{\mathbb{R}^{N}}\int _{\mathbb{R}^{N}}\frac {[u(x)-u(z)][v(x)-v(z)]}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x)u(x)v(x)\,dx\, + \\ &{}-\int_{\mathbb{R}^{N}}f \bigl(x,u(x) \bigr)v(x)\,dx-\lambda\int _{\mathbb {R}^{N}}h(x)|u|^{p-2}uv\,dx \end{aligned}$$
(2.3)

for all \(u,v\in X\). In addition, any critical point of J on X is a solution of problem (1.1).

Next, we give the variant version of the mountain pass theorem which is important for the proof of our main results.

Theorem 2.1

(see [21])

Let E be a real Banach space with its dual space \(E^{\ast}\), and suppose that \(I\in C^{1}(E,\mathbb{R})\) satisfies

$$\max \bigl\{ I(0),I(e) \bigr\} \leq\mu< \eta\leq\inf_{\|u\|=\rho}I(u) $$

for some \(\mu<\eta\), \(\rho>0\) and \(e\in E\) with \(\|e\|>\rho\). Let \(\hat {c}\geq\eta\) be characterized by

$$\hat{c}=\inf_{\gamma\in\Gamma}\max_{0\leq\tau\leq 1}I \bigl(\gamma( \tau) \bigr), $$

where \(\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)=e\}\) is the set of continuous paths joining 0 and e, then there exists a sequence \(\{u_{n}\}\subset E\) such that

$$I(u_{n})\rightarrow\hat{c}\geq\eta \quad\textit{and}\quad \bigl(1+ \|u_{n}\| \bigr)\bigl\| I'(u_{n})\bigr\| _{E^{\ast}} \rightarrow0, \quad\textit{as } n \rightarrow\infty. $$

3 Proof of the main results

To prove our main results, we first give the following lemma.

Lemma 3.1

For any real number \(2_{\alpha}^{\ast}-1>k\geq 1\), assume that the conditions (V0), (F1)-(F2) hold. Then there exists \(\Lambda>0\) such that for every \(\lambda\in(0,\Lambda)\) there are two positive constants ρ, η such that \(J(u)|_{\|u\|=\rho}\geq\eta>0\).

Proof

For any \(\epsilon>0\), it follows from the conditions (F1)-(F2) that there exist \(C_{\epsilon}>0\) and \(2_{\alpha }^{\ast}>r>\max\{2,k\}\) such that

$$ F(x,s)\leq\frac{|b|_{\infty}+\epsilon}{2}s^{2}+\frac{C_{\epsilon }}{r}|s|^{r}, \quad \forall s\in\mathbb{R}. $$
(3.1)

By (2.2) and (3.1), Sobolev’s inequality, and Hölder’s inequality, one has

$$\begin{aligned} J(u)&= \frac{1}{2}\|u\|^{2}-\int _{\mathbb {R}^{N}}F(x,u)\,dx-\frac{\lambda}{p}\int_{\mathbb{R}^{N}}h(x)|u|^{p}\,dx \\ &\geq\frac{1}{2}\|u\|^{2}-\int_{\mathbb{R}^{N}} \frac{|b|_{\infty }+\epsilon}{2}u(x)^{2}\,dx-\int_{\mathbb{R}^{N}} \frac{C_{\epsilon }}{r}u(x)^{r}\,dx-\frac{\lambda}{p}\int _{\mathbb{R}^{N}}h(x)\bigl|u(x)\bigr|^{p}\,dx \\ &\geq\frac{1}{2}\|u\|^{2}-\frac{|b|_{\infty}+\epsilon}{2S_{2}^{2}}\|u\| ^{2}-\frac{C_{\epsilon}}{rS_{r}^{r}}\|u\|^{r}-\frac{\lambda S_{2}^{-p}}{p}\|h \|_{\frac{2}{2-p}}\|u\|^{p} \\ &=\|u\|^{p} \biggl[\frac{1}{2} \biggl(1-\frac{|b|_{\infty}+\epsilon}{2S_{2}^{2}} \biggr) \|u\| ^{2-p}-\frac{C_{\epsilon}}{rS_{r}^{r}}\|u\|^{r-p}- \frac{\lambda S_{2}^{-p}}{p}\|h \|_{\frac{2}{2-p}} \biggr] \end{aligned}$$
(3.2)

for all \(u\in X\). Take \(\epsilon=\frac{S_{2}^{2}}{2}-|b|_{\infty}\) and define

$$l(t)=\frac{1}{4}t^{2-p}-C_{\epsilon}r^{-1}S_{r}^{-r}t^{r-p}, \quad \forall t\geq0. $$

It is easy to prove that there exists \(\rho>0\) such that

$$\max_{t\geq0}l(t)=l(\rho)=\frac{r-2}{4(r-p)} \biggl[ \frac {(2-p)rS_{r}^{r}}{4(r-p)C_{\epsilon}} \biggr]^{\frac{2-p}{r-2}}. $$

Then it follows from (3.2) that there exists \(\Lambda>0\) such that for every \(\lambda\in(0,\Lambda)\) there exist two positive constants ρ, η such that \(J(u)|_{\|u\|=\rho}\geq\eta>0\). □

Consider the minimum problem

$$ \begin{aligned}[b] \mu={}&\inf \biggl\{ \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac {|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x)u(x)^{2}\,dx: u\in H^{\alpha} \bigl({\mathbb{R}^{N}} \bigr),\\ &{}\int_{\mathbb {R}^{N}}q(x)u(x)^{2}\,dx=1 \biggr\} . \end{aligned} $$
(3.3)

Then we have the following results.

Lemma 3.2

There exist a constant \(c_{1}>0\) and \(\phi _{1}\in H^{\alpha}(\mathbb{R}^{N})\) with \(\int_{\mathbb {R}^{N}}q(x)\phi_{1}(x)^{2}\,dx=1\) such that \(\mu\geq c_{1}\) and

$$ \mu=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|\phi_{1}(x)-\phi _{1}(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz + \int_{\mathbb{R}^{N}}V(x)\phi _{1}(x)^{2}\,dx, $$
(3.4)

i.e. the minimum (3.3) is achieved.

Proof

For any \(u\in H^{\alpha}(\mathbb{R}^{N})\) with \(\int_{\mathbb{R}^{N}}q(x)u(x)^{2}\,dx=1\), by Lemma 2.1 and Sobolev’s embedded theorem, we have

$$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac {|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+ \int_{\mathbb {R}^{N}}V(x)u(x)^{2}\,dx\geq c_{0}\|u \|^{2}_{H}\geq c_{0}\|u\| _{2}^{2} \geq\frac{c_{0}}{|q|_{\infty}}>0. $$

Therefore, there exists a constant \(c_{1}>0\) such that \(\mu\geq c_{1}\). Let \(\{u_{n}\}\subset H^{\alpha}({\mathbb{R}^{N}})\) be a minimizing sequence of (3.3). Clearly, \(\int_{\mathbb{R}^{N}}q(x)u_{n}(x)^{2}\,dx=1\) and \(\{u_{n}\}\) is bounded. Then there exist a subsequence \(\{u_{n}\}\) and \(\phi_{1}\in H^{\alpha}({\mathbb{R}^{N}})\) such that \(u_{n}\rightharpoonup\phi_{1}\) weakly in \(H^{\alpha}({\mathbb {R}^{N}})\) and \(u_{n}\rightarrow\phi_{1}\) strongly in \(L^{2}(\mathbb {R}^{N})\). So it is easy to verify that \(\int_{\mathbb {R}^{N}}q(x)u_{n}(x)^{2}\,dx\rightarrow\int_{\mathbb{R}^{N}}q(x)\phi _{1}(x)^{2}\,dx\) as \(n\rightarrow\infty\) and \(\int_{\mathbb {R}^{N}}q(x)\phi_{1}(x)^{2}\,dx=1\). Therefore,

$$\begin{aligned} \mu&\leq \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|\phi _{1}(x)-\phi_{1}(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x)\phi_{1}(x)^{2}\,dx \\ &\leq\liminf_{n\rightarrow\infty} \biggl\{ \int_{\mathbb{R}^{N}}\int _{\mathbb{R}^{N}}\frac{|u(x)-u(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz +\int_{\mathbb{R}^{N}}V(x)u(x)^{2}\,dx \biggr\} \\ &\leq\mu. \end{aligned}$$
(3.5)

This implies that

$$\mu=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|\phi_{1}(x)-\phi _{1}(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb{R}^{N}}V(x)\phi _{1}(x)^{2}\,dx. $$

 □

Lemma 3.3

For any real number \(2_{\alpha}^{\ast}-1>k\geq 1\), assume that the conditions (V0), (F1)-(F2) hold. Let \(\rho,\Lambda>0\) be as in Lemma  3.1. Then we have the following results:

  1. (i)

    If \(k=1\) and \(\mu<1\), then there exists \(e\in X\) with \(\|e\| >\rho\) such that \(J(e)<0 \) for all \(\lambda\in(0,\Lambda)\).

  2. (ii)

    If \(k>1\), then there exists \(e\in X\) with \(\|e\|>\rho\) such that \(J(e)<0 \) for all \(\lambda\in(0,\Lambda)\).

Proof

(i) In case \(k=1\). Since \(\mu<1\), we can choose a nonnegative function \(\varphi\in H^{\alpha}({\mathbb{R}^{N}})\) with \(\int_{\mathbb{R}^{N}}q(x)\varphi(x)^{2}\,dx=1\) such that

$$\|\varphi\|^{2}=\int_{\mathbb{R}^{N}}\int _{\mathbb{R}^{N}}\frac{|\varphi (x)-\varphi(z)|^{2}}{|x-z|^{N+2\alpha}}\,dx\,dz+\int_{\mathbb {R}^{N}}V(x) \varphi(x)^{2}\,dx< 1. $$

Therefore, by the condition (F2) and Fatou’s lemma, we have

$$\begin{aligned} \lim_{t\rightarrow+\infty}\frac{J(t\varphi )}{t^{2}}&\leq \frac{1}{2}\|\varphi\|^{2}-\lim_{t\rightarrow +\infty}\int _{\mathbb{R}^{N}}\frac{F(x,t\varphi)}{t^{2}\varphi ^{2}}\varphi^{2}\,dx-\lim _{t\rightarrow+\infty}\frac{\lambda }{pt^{2-p}}\int_{\mathbb{R}^{N}}h(x)| \varphi|^{p}\,dx \\ &\leq\frac{1}{2}\|\varphi\|^{2}-\frac{1}{2}\int _{\mathbb {R}^{N}}q(x)\varphi^{2}\,dx \\ &=\frac{1}{2} \bigl(\|\varphi\|^{2}-1 \bigr)< 0. \end{aligned}$$
(3.6)

So, \(J(t\varphi)\rightarrow-\infty\) as \(t\rightarrow+\infty\), then there exists \(e\in X\) with \(\|e\|>\rho\) such that \(J(e)<0 \) for all \(\lambda\in(0,\Lambda)\).

(ii) In case \(k>1\). \(q\in L^{\infty}(\mathbb{R}^{N},\mathbb {R}^{+})\) with \(q^{+}\neq0\), we can choose a function \(\omega\in H^{\alpha}({\mathbb{R}^{N}})\) such that

$$\int_{\mathbb{R}^{N}}q(x)|\omega|^{k+1}\,dx>0. $$

Therefore, by the condition (F2) and Fatou’s lemma, we have

$$\begin{aligned} \lim_{t\rightarrow+\infty}\frac{J(t\omega)}{t^{k+1}} &\leq \frac{\|\omega\|^{2}}{2t^{k-1}}-\lim_{t\rightarrow+\infty }\int_{\Omega} \frac{F(x,t\omega)}{t^{k+1}\omega^{k+1}}\varphi ^{k+1}\,dx-\lim_{t\rightarrow+\infty} \frac{\lambda }{pt^{k+1-p}}\int_{\Omega}\xi(x)|\omega|^{p}\,dx \\ &\leq-\frac{1}{k+1}\int_{\Omega}q(x)\omega^{k+1}\,dx \\ &< 0. \end{aligned}$$
(3.7)

So, \(J(t\omega)\rightarrow-\infty\) as \(t\rightarrow+\infty\); then there exists \(e\in X\) with \(\|e\|>\rho\) such that \(J(e)<0 \) for all \(\lambda\in(0,\Lambda)\). □

Next, we define

$$\beta=\inf_{\gamma\in\Gamma}\max_{0\leq\tau\leq1}J \bigl(\gamma ( \tau) \bigr), $$

where \(\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)=e\}\). Then by Theorem 2.1, Lemma 3.1, and Lemma 3.3, there exists a sequence \(\{ u_{n}\}\subset X\) such that

$$ J(u_{n})\rightarrow\beta \quad\mbox{and}\quad \bigl(1+\|u_{n} \|\bigr)\bigl\| J'(u_{n})\bigr\| _{E^{\ast}}\rightarrow0, \quad \mbox{as } n \rightarrow \infty. $$
(3.8)

Then we have the following results.

Lemma 3.4

For any real number \(2_{\alpha}^{\ast}-1>k\geq 1\), assume that the conditions (V0), (F1)-(F3) hold. Let \(\Lambda>0\) be as in Lemma  3.1. Then \(\{u_{n}\}\) defined by (3.8) is bounded in X for all \(\lambda\in(0,\Lambda)\).

Proof

For n large enough, by Hölder’s inequality and Lemma 2.1, one has

$$\begin{aligned} \beta+1\geq{}& J(u_{n})-\frac{1}{\theta} \bigl\langle J'(u_{n}),u_{n} \bigr\rangle \\ ={}& \biggl(\frac{1}{2}-\frac{1}{\theta} \biggr)\|u_{n} \|^{2}-\int_{\mathbb {R}^{N}} \biggl[F(x,u_{n})- \frac{1}{\theta}f(x,u_{n})u_{n} \biggr]\,dx \\ &{}-\lambda \biggl(\frac{1}{p}-\frac{1}{\theta} \biggr)\int _{\mathbb {R}^{N}}h(x)|u_{n}|^{p}\,dx \\ \geq{}& \frac{\theta-2}{2\theta}\|u_{n}\|^{2}-d_{0} \int_{\mathbb {R}^{N}}u_{n}^{2}\,dx -\lambda \biggl( \frac{1}{p}-\frac{1}{\theta} \biggr)\int_{\mathbb {R}^{N}}h(x)|u_{n}|^{p}\,dx \\ \geq{}& \frac{\theta-2}{2\theta}\|u_{n}\|^{2}-\frac{d_{0}}{S_{2}^{2}} \| u_{n}\|^{2} -\lambda \biggl(\frac{1}{p}- \frac{1}{\theta} \biggr)S_{2}^{-p}\|h\|_{\frac {2}{2-p}} \|u_{n}\|^{p} \\ \geq{}& \biggl(\frac{\theta-2}{2\theta}-\frac{d_{0}}{S_{2}^{2}} \biggr)\|u_{n} \|^{2} -\Lambda \biggl(\frac{1}{p}-\frac{1}{\theta} \biggr)S_{2}^{-p}\|h\|_{\frac{2}{2-p}}\| u_{n} \|^{p}, \end{aligned}$$
(3.9)

which implies that \(\{u_{n}\}\) is bounded in X, since \(1\leq p<2\). □

Denote \(B_{\rho}=\{u\in X:\|u\|<\rho\}\), where ρ is given by Lemma 3.1. Then by Ekeland’s variational principle and Lemma 2.2, we have the following lemma, which shows that J has a local minimum if λ is small.

Lemma 3.5

For any real number \(2_{\alpha}^{\ast}-1>k\geq 1\), assume that the conditions (V0), (F1)-(F3). Let \(\Lambda>0\) be as in Lemma  3.1. Then for every \(\lambda\in(0,\Lambda )\), there exists \(u_{0}\in X\) such that

$$J(u_{0})=\inf \bigl\{ J(u):u\in\bar{B}_{\rho} \bigr\} < 0, $$

and \(u_{0}\) is a solution of problem (1.1).

Proof

Since \(h\in L^{\frac{2}{2-p}}\setminus\{0\}\) with \(h^{+}\neq0\), we can choose a function \(\psi\in H^{\alpha}({\mathbb{R}^{N}})\) such that

$$\int_{\mathbb{R}^{N}}h(x)|\psi|^{p}\,dx>0. $$

Hence, we have

$$\begin{aligned} J(t\psi)&= \frac{t^{2}}{2}\|\psi\|^{2}-\int _{\mathbb{R}^{N}}F(x,t\psi)\,dx-\frac{\lambda t^{p}}{p}\int_{\mathbb {R}^{N}}h(x)| \psi|^{p}\,dx \\ &\leq \frac{t^{2}}{2}\|\psi\|^{2}-\frac{\lambda t^{p}}{p}\int _{\mathbb {R}^{N}}h(x)|\psi|^{p}\,dx< 0 \end{aligned}$$
(3.10)

for \(t>0\) small enough, which implies \(\theta_{0}:=\inf\{J(u):u\in\bar {B}_{\rho}\}<0\). By Ekeland’s variational principle, there exists a minimizing sequence \(\{u_{n}\}\subset\bar{B}_{\rho}\) such that \(J(u_{n})\rightarrow\theta_{0}\) and \(J'(u_{n})\rightarrow0\) as \(n\rightarrow\infty\). Hence Lemma 2.2 implies that there exists \(u_{0}\in X\) such that \(J'(u_{0})=0\) and \(J(u_{0})=c_{1}<0\). □

Proof of Theorem 1.1

From Lemma 2.2 and Lemma 3.4, there is a \(\bar{u}\in X\) such that, up to a subsequence, \(u_{n}\rightharpoonup\bar{u}\) weakly in X, \(u_{n}\rightarrow\bar {u}\) strongly in \(L^{s}(\mathbb{R})\) for \(s\in[2,2_{\alpha}^{\ast})\). By using a standard procedure, we can prove that \(u_{n}\rightarrow\bar {u}\) strongly in X. Moreover, \(J(\bar{u})=\beta>0\) and \(\bar{u}\) is another solution of problem (1.1). Thus, combining with Lemma 3.5, we prove that problem (1.1) has at least two solutions \(u_{0},\bar{u}\in X\) satisfying \(J(u_{0})<0\) and \(J(\bar{u})>0\). □

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Barrios, B, Colorado, E, de Pablo, A, Sánchez, U: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133-6162 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brändle, C, Colorado, E, de Pablo, A: A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb., Sect. A 143, 39-71 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Di Blasio, G, Volzone, B: Comparison and regularity results for the fractional Laplacian via symmetrization methods. J. Differ. Equ. 253, 2593-2615 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Servadei, R, Valdinoci, E: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887-898 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Yu, X: The Nehari manifold for elliptic equation involving the square root of the Laplacian. J. Differ. Equ. 252, 1283-1308 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Silvestre, L: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67-112 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Tan, J, Wang, Y, Yang, J: Nonlinear fractional field equations. Nonlinear Anal. 75, 2098-2110 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L: Surfaces minimizing nonlocal energies. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 20, 281-299 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caffarelli, L, Vazquez, JL: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537-565 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Laskin, N: Fractional quantum mechanics and path integrals. Phys. Lett. A 269, 298-305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, X, Wang, Z: Ground state solutions of scalar field equations involving fractional Laplacian with general nonlinearity. Nonlinearity 26, 479-494 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Feng, B: Ground state solutions for the fractional equation. Electron. J. Differ. Equ. 2013, 127 (2013)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Secchi, S: Ground state solutions for the fractional equation in \(\mathbb{R}^{N}\). J. Math. Phys. 54, 031501 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dipierro, S, Palatucci, G, Valdinoci, E: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201-216 (2013)

    MATH  MathSciNet  Google Scholar 

  17. Autuori, G, Pucci, P: Elliptic problems involving the fractional Laplacian in \(\mathbb{R}^{N}\). J. Differ. Equ. 255, 2340-2362 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Secchi, S: On fractional Schrödinger equations in \(\mathbb{R}^{N}\) without the Ambrosetti-Rabinowitz condition. arXiv:1210.0755

  19. Torres, C: Non-homogeneous fractional Schrödinger equation. arXiv:1311.0708

  20. Chang, X: Ground state solutions of asymptotically linear fractional Schrödinger equation. J. Math. Phys. 54, 061504 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ekeland, I: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline in Hunan Province.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Liu Yang.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, L. Multiplicity of solutions for fractional Schrödinger equations with perturbation. Bound Value Probl 2015, 56 (2015). https://doi.org/10.1186/s13661-015-0317-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-015-0317-5

MSC

Keywords