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Journal of Function Spaces
Volume 2019, Article ID 8981528, 8 pages
https://doi.org/10.1155/2019/8981528
Research Article

Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

1School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
2School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

Correspondence should be addressed to Anran Li; nc.ude.uxs@narnail

Received 8 June 2019; Accepted 16 July 2019; Published 31 July 2019

Academic Editor: Richard I. Avery

Copyright © 2019 Lizhen Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to . These results extend some known results in previous papers.

1. Introduction

We consider the following system via variational methods:where , , , . and represent the Laplace operator of the fractional order. If , then the system degenerates into the standard Schrödinger-Poisson system, which describes the interaction between the same charged particles when the magnetic effect can be ignored [1]. In recent years, the existence, multiplicity, and centralization of solutions for the Schrödinger-Poisson system have been deeply studied via variational methods, and a great number of works have been obtained, see, for example, [28]. On the other hand, is a class of nonlocal pseudo-differential operators. Since nonlocal differential equations can better and more fully describe the physical experimental phenomena than classical local differential operators, the study of nonlinear fractional Laplace equation has become one of the most popular research fields in nonlinear analysis.

In the literature [9], Wei considered the following system:By using the critical point theory, the author obtained infinitely many solutions when . In the literature [10], Teng studied a system of the formwhere , , . In [11], Zhang, Marcos, and Squassina used perturbation approach to obtain the existence of solutions for the following system when the nonlinear term is subcritical or criticalwhere , . In [12], Duarte and Souto investigated the following system via variational methodswhere , , is a periodic potential. A positive solution and a ground state solution were got in [12]. In [13], Li studied a system of the following form:where , . Combining the perturbation method with mountain pass theorem, the existence of nontrivial solutions was obtained in [13]. In [14], Yu, Zhao, and Zhao studied the following fractional Schrödinger–Poisson system with critical growth via variational methodswhere , the potential is continuous with positive infimum, is continuous and subcritical at infinity. Under some Monotone hypothesis on , the existence of positive ground state solution is got in [14]. For small , a multiple result is also got in [14].

Inspired by [916], in this paper, we prove the existence of multiple solutions for system by symmetric mountain pass theorem. Moreover, we prove the system has a sequence of negative energy solutions by dual fountain theorem. The assumptions on and nonlinearity in this paper are given below:

(V) , and ;

(H1) , and there exists such that , where , ;

(H2) there exist and such that , for . Moreover, ;

(H3) , , ;

(H4) , and .

2. Preliminaries

For , denotes the usual Lebesgue space with norm . Fix , fractional Sobolev space denoted asequipped with the normwhere denotes the Fourier transform of function . Let ; the fractional Laplacian operator is defined byAccording to Plancherel theorem [17], one has , . By (8), we define the equivalent norm is denoted asIn particular, is the completion of , with respect to the norm, represents a weighted Lebesgue space, that is,equipped with the norm

For convenience, we use to represent any positive constants which may change from line to line. According to [18], the embedding is continuous for all , i.e., there exists satisfyingSo, the embedding is continuous when . Fix ; we define the nonlinear operator byHenceBy Lax-Milgram theorem, we can find a unique such thatand is expressed asAccording to (19), for all . Since , , we can also get . Together with (17) and (18),From Hölder’s inequality and (19),Evidently,Substituting (18) into system , system is equivalent to

For the equation , we define the work space aswith

Lemma 1 (see [19]). Assume condition (V) holds, , ; then the embedding is compact.

From (V) and (H1)-(H4), is well defined. Moreover, by Lemma 1, with

Proposition 2. (i) If equation has a solution , then system has a solution .
(ii) If for every , the following equationholds, then is a solution of .

Set as a set of normalized orthogonal basis of , . . Obviously, .

Definition 3 (see [20, 21]). Set , . If any sequence satisfyinghas a convergent subsequence, then satisfies the condition. Any sequence satisfying (28) is called the sequence.

Definition 4 (see [21, 22]). Set , . If every sequence , satisfyinghas a convergent subsequence, then satisfies the condition.

Proposition 5 (see [23]). Let be a Banach space with . Assume is an even functional and satisfies the condition and
(A1) there exist , satisfying ;
(A2) for every linear subspace with , there exists a constant such that ,
then has a list of unbounded critical points.

Proposition 6 (see [22]). Assume that is an even functional, . If for any , there exist such that(C1);(C2);(C3), ;(C4)for every , satisfies the condition, then has a sequence of negative critical points that converge to .

3. Main Results

Lemma 7. If hypotheses (V) and (H1)-(H4) hold, then for any , satisfies the condition.

Proof. First, we prove the sequence of is bounded. According to (H4), it is easy to get thatBy condition (H2), there exists such thatMoreover, for any given , we can choose a constant such thatFrom condition (H1), when , one hasSo, for any ,For , there exists such thatNow (34) impliesSince is the sequence, when is large enough,Thus, according to (36), when is large enough,Therefore, is a bounded sequence in . By Lemma 1, there exists such thatNext, we define the linear operator asFrom Hölder’s inequality, we obtainNow by Lemma 1 and (22),Similarly, we can also proveSince in ,
At last, combining Hölder’s inequality with (H1) and (H4), we can easily getThusthat is,

Lemma 8. If hypotheses (V) and (H1)-(H4) hold, then satisfies condition for all .

Proof. By Definition 4, we just prove the following fact: if for any , , and , as , then has a convergence subsequence. The proof method is similar to Lemma 7.

Lemma 9 (see [24]). For , , setand then , .

Theorem 10. If hypotheses (V) and (H1)-(H4) hold, then we can find , such that system (SP) has multiple solutions for every . Moreover, the corresponding energy values tend to infinity.

Proof. According to Lemma 7, satisfies condition. We only need to prove that satisfies (A1) and (A2). By virtue of (H1),From Lemma 9, we can getTake a sufficiently large such that . Combining the above inequality, we obtainSetSince , there existssuch that . Therefore, ,On the other hand, by conditions (H1) and (H2), there exist positive constants such thatSince all the norms are equivalent in every finite linear subspace , then for For , as . Then there exists such that . Thus, according to Proposition 5, the system (SP) has a list of solutions , and the corresponding energy values tend to infinity.

Theorem 11. If hypotheses (V) and (H1)-(H4) hold, then the system (SP) has a sequence of negative energy solutions for all , and the energy values tend to .

Proof. By Lemma 8, for all , satisfies the condition. It now remains to show that (C1)-(C3) are satisfied. According to Lemma 9, for every , , as . Thus there exists such that for . For , there exists such thatHence, for with , it follows thatFor every , let . By Lemma 9, , as . Thus, there exists , such that for every , , for with .
Secondly, since for every fixed , the norms are equivalent in , when is sufficiently large, there exists a small enough such that and for with .
Finally, according to (C3), when , for , with , one hasSince , , as , therefore (C3) holds. By Proposition 6, has a list of solutions such that

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11701346, 11571209).

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