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.
Keywords:Schrödinger-Maxwell equations; sublinear; minimizing theorem; dual fountain theorem
1 Introduction and main result
Consider the following semilinear Schrödinger-Maxwell equations:
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 .
Assume that the following conditions hold:
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 1.2Assume thatVsatisfies
andfsatisfies the following conditions.
By Theorem 1.2, we obtain the following corollary.
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.
Remark 1.6 In Theorem 1.5 we obtain the existence of solutions for problem (1) under the assumption that is indefinite and without any coercive assumptions respect to V such as (). There are functions V and f which satisfy Theorem 1.5, but do not satisfy the corresponding results in [2-16]. For example,
2 Preliminary results
In order to establish our results via critical point theory, we firstly describe some properties of the space , on which the variational functional associated with problem (1) is defined. Define the function space
equipped with the norm
and the function space
with the norm
equipped with the inner product
and the corresponding norm
Note that the following embeddings
It is clear that
Similarly, we can prove
It follows from (15), (16), (17) and (18) that
which together with Lebesgue’s convergence theorem shows
In the proof of Theorem 1.2, the following lemma is needed.
We obtain the existence of a solution for problem (1) by using the following standard minimizing argument.
is a critical value of Φ.
contains a subsequence converging to a critical point of Φ.
Now we show the following dual fountain theorem.
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.
By the Hölder inequality and using the second inequality in (5), we have
It follows from (23) that
and by the Hölder inequality, we have
and it follows that
By (12), we have
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, .
Then by Lebesgue’s convergence theorem, we have
(i) By Lemma 2.4
Then we have
Therefore, we have
The author declares that she has no competing interests.
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).
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