Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations
© Lv; licensee Springer 2013
Received: 24 September 2012
Accepted: 5 July 2013
Published: 30 July 2013
In this paper I consider a class of sublinear Schrödinger-Maxwell equations, and new results about the existence and multiplicity of solutions are obtained by using the minimizing theorem and the dual fountain theorem respectively.
1 Introduction and main result
Such a system, also known as the nonlinear Schrödinger-Poisson system, arises in an interesting physical context. Indeed, according to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger and the Maxwell equations (we refer to [1, 2] for more details on the physical aspects and on the qualitative properties of the solutions). In particular, if we are looking for electrostatic-type solutions, we just have to solve (1).
In recent years, system (1), with or being radially symmetric, has been widely studied under various conditions on f; see, for example, [3–11]. Since (1) is set on , it is well known that the Sobolev embedding () is not compact, and then it is usually difficult to prove that a minimizing sequence or a sequence that satisfies the condition, briefly a Palais-Smale sequence, is strongly convergent if we seek solutions of (1) by variational methods. If is radial (for example, ), we can avoid the lack of compactness of Sobolev embedding by looking for solutions of (1) in the subspace of radial functions of , which is usually denoted by , since the embedding () is compact. Specially, Ruiz  dealt with (1) under the assumption that and () and got some general existence, nonexistence and multiplicity results.
Moreover, in  the authors considered system (1) with periodic potential , and the existence of infinitely many geometrically distinct solutions was proved by the nonlinear superposition principle established in .
There are also some papers treating the case with nonradial potential . More precisely, Wang and Zhou  got the existence and nonexistence results of (1) when is asymptotically linear at infinity. Chen and Tang  proved that (1) has infinitely many high energy solutions under the condition that is superlinear at infinity in u by the fountain theorem. Soon after, Li, Su and Wei  improved their results.
Up to now, there have been few works concerning the case that is nonradial potential and is sublinear at infinity in u. Very recently, Sun  treated the above case based on the variant fountain theorem established in Zou .
Theorem 1.1 
Assume that the following conditions hold:
() satisfies , where is a constant. For every , .
() , where , is a positive function such that and .
In the present paper, based on the dual fountain theorem, we can prove the same result under a more generic condition, which generalizes the result in . Our first result can be stated as follows.
Theorem 1.2 Assume that V satisfies
() and ;
and f satisfies the following conditions.
for all and ;
for all and ;
for all and ;
for all and , where , ;
() for all and .
By Theorem 1.2, we obtain the following corollary.
Corollary 1.3 Assume that L satisfies () and W satisfies
() , where , is a constant and is a function such that and for , where .
Remark 1.4 In Theorem 1.2, infinitely many solutions for problem (1) are obtained under the symmetry condition () by using the dual fountain theorem. As a special case of Theorem 1.2, Corollary 1.3 generalizes and improves Theorem 1.1. To show this, it suffices to compare () and (), () and (). Firstly, it is clear that () is really weaker than (). Secondly, in () a is assumed to be positive, while in () we assume that a is indefinite.
Moreover, under all the conditions of Theorem 1.2 except () we obtain an existence result.
Theorem 1.5 Assume that L satisfies () and W satisfies (), (), (), (). Then problem (1) possesses a nontrivial solution.
2 Preliminary results
Lemma 2.1 Suppose that assumption () holds. Then the embedding of E in is compact, where , is positive for a.e. .
Now it follows from (6) and (7) that K is precompact in . Obviously, we have E is compact embedded in , where , is positive for a.e. . □
as . Now we have proved the lemma. □
In the proof of Theorem 1.2, the following lemma is needed.
Lemma 2.3 Assume that is an open set. Then, for any closed set , there exists a function such that for all , for all and for all .
then for all and for all . Moreover, by the definition of , we have and . □
Since E is a Hilbert space, then there exists a basis such that , where . Letting , , now we show the following lemma, which will be used in the proof of Theorem 1.2.
in for all and , which together with (20) and (21) implies that for all and . □
We obtain the existence of a solution for problem (1) by using the following standard minimizing argument.
Lemma 2.5 
is a critical value of Φ.
In order to prove the multiplicity of solutions, we will use the dual fountain theorem. Firstly, we introduce the definition of the condition.
contains a subsequence converging to a critical point of Φ.
Now we show the following dual fountain theorem.
Lemma 2.7 
Moreover, if satisfies the condition for all , then Φ has a sequence of critical points such that as .
3 Proof of theorems
It is easy to know that I exhibits a strong indefiniteness, namely it is unbounded both from below and from above on an infinitely dimensional subspace. This indefiniteness can be removed using the reduction method described in , by which we are led to study a variable functional that does not present such a strong indefinite nature.
It can be proved that is a solution of problem (1) if and only if is a critical point of the functional Φ and ; see, for instance, .
Lemma 3.1 Under conditions (), (), (), (), Φ satisfies the condition.
for all .
Noting that for all , so is bounded.
Consequently, as . Φ satisfies the condition. □
for all . The rest of the proof is the same as that of Lemma 3.1.
For every (), by Lemma 2.3 there exists such that and . Letting , can be extended to be a basis . Therefore , where . Now we define , .
- (i)By Lemma 2.4
- (ii)For any , there exists such that
- (iii)By (40), for any with , we have
as . □
when ε and λ are both small enough. Since , then . Hence, is a nontrivial solution of problem (1). □
The author is highly grateful for the referees’ careful reading and comments on this paper. This work is partially supported by the National Natural Science Foundation of China (No. 11071198) and Southwest University Doctoral Fund Project (SWU112107).
- Benci V, Fortunato D: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 1998, 11: 283-293.MathSciNetGoogle Scholar
- Polidoro S, Ragusa MA: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 2008, 3(24):1011-1046.MathSciNetView ArticleGoogle Scholar
- Azzollini A, Pomponio A: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 2008, 345: 90-108. 10.1016/j.jmaa.2008.03.057MathSciNetView ArticleGoogle Scholar
- Coclite GM: A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun. Appl. Anal. 2003, 7(2-3):417-423.MathSciNetGoogle Scholar
- Coclite GM: A multiplicity result for the Schrödinger-Maxwell equations with negative potential. Ann. Pol. Math. 2002, 79(1):21-30. 10.4064/ap79-1-2MathSciNetView ArticleGoogle Scholar
- D’Aprile T, Mugnai D: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. R. Soc. Edinb. A 2004, 134: 893-906. 10.1017/S030821050000353XMathSciNetView ArticleGoogle Scholar
- Zhao L, Zhao F: Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal. 2009, 70: 2150-2164. 10.1016/j.na.2008.02.116MathSciNetView ArticleGoogle Scholar
- Kikuchi H: On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal. 2007, 67(5):1445-1456. 10.1016/j.na.2006.07.029MathSciNetView ArticleGoogle Scholar
- Salvatore A:Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in . Adv. Nonlinear Stud. 2006, 6(2):157-169.MathSciNetGoogle Scholar
- Ambrosetti A, Ruiz D: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 2008, 10: 391-404. 10.1142/S021919970800282XMathSciNetView ArticleGoogle Scholar
- Ruiz D: The Schrödinger-Maxwell equation under the effect of a nonlinear local term. J. Funct. Anal. 2006, 237: 655-674. 10.1016/j.jfa.2006.04.005MathSciNetView ArticleGoogle Scholar
- Zhao L, Zhao F: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl. 2008, 346: 155-169. 10.1016/j.jmaa.2008.04.053MathSciNetView ArticleGoogle Scholar
- Ackermann N: A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J. Funct. Anal. 2006, 234: 277-320. 10.1016/j.jfa.2005.11.010MathSciNetView ArticleGoogle Scholar
- Wang Z, Zhou H:Positive solution for a nonlinear stationary Schrödinger-Poisson system in . Discrete Contin. Dyn. Syst. 2007, 18: 809-816.MathSciNetView ArticleGoogle Scholar
- Chen S, Tang C: High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal. 2009, 71: 4927-4934. 10.1016/j.na.2009.03.050MathSciNetView ArticleGoogle Scholar
- Li Q, Su H, Wei Z: Existence of infinitely many large solutions for the nonlinear Schrödinger-Maxwell equations. Nonlinear Anal. 2010, 72: 4264-4270. 10.1016/j.na.2010.02.002MathSciNetView ArticleGoogle Scholar
- Sun J: Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equation. J. Math. Anal. Appl. 2012, 390: 514-522. 10.1016/j.jmaa.2012.01.057MathSciNetView ArticleGoogle Scholar
- Zou W: Variant fountain theorems and their applications. Manuscr. Math. 2001, 104: 343-358. 10.1007/s002290170032View ArticleGoogle Scholar
- Rabinowitz PH CBMS Regional Conference Series in Mathematics 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
- Willem W: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar
- Evans LC: Partial Differential Equations. Am. Math. Soc., Providence; 1998.Google Scholar
- Benmlih K:Stationary solutions for a Schrödinger-Poisson system in . Electron. J. Differ. Equ. Conf. 9. Proceedings of the 2002 Fez Conference on Partial Differential Equations 2002, 65-76.Google Scholar
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