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# Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations

Ying Lv

Author Affiliations

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

Boundary Value Problems 2013, 2013:177  doi:10.1186/1687-2770-2013-177

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/177

 Received: 24 September 2012 Accepted: 5 July 2013 Published: 30 July 2013

© 2013 Lv; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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:

(1)

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 [11] dealt with (1) under the assumption that and () and got some general existence, nonexistence and multiplicity results.

Moreover, in [12] 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 [13].

There are also some papers treating the case with nonradial potential . More precisely, Wang and Zhou [14] got the existence and nonexistence results of (1) when is asymptotically linear at infinity. Chen and Tang [15] 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 [16] 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 [17] treated the above case based on the variant fountain theorem established in Zou [18].

Theorem 1.1[17]

Assume that the following conditions hold:

() satisfies, whereis a constant. For every, .

() , where, is a positive function such thatand.

Then problem (1) has infinitely many nontrivial solutionssatisfying

as.

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 [17]. Our first result can be stated as follows.

Theorem 1.2Assume thatVsatisfies

() and;

andfsatisfies the following conditions.

() There exist constants, and a functionsuch that

for alland;

() There exist constants, and a functionsuch that

for alland;

() For every, there exist a constantand a functionsuch that

for alland;

() There exist constants, andsuch that

for alland, where, ;

() for alland.

Then problem (1) has infinitely many nontrivial solutionssatisfying

as.

By Theorem 1.2, we obtain the following corollary.

Corollary 1.3Assume thatLsatisfies () andWsatisfies

() , where, is a constant andis a function such thatandfor, where.

Then problem (1) has infinitely many nontrivial solutionssatisfying

as.

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 1.5Assume thatLsatisfies () andWsatisfies (), (), (), (). Then problem (1) possesses a nontrivial solution.

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)

and

(3)

in which . It is clear that is indefinite. Denoting by π the area of the unit ball in , we obtain

(4)

which means that . So, (2) satisfies our results, but does not satisfy the results in [3-17].

### 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

Let

equipped with the inner product

and the corresponding norm

Note that the following embeddings

are continuous, where is the critical exponent for the Sobolev embeddings in dimension 3. Therefore, there exist constants and such that

(5)

for all . Here () denotes the Banach spaces of a function on with values in R under the norm

Let

where for a.e. . Then is a Banach space with the norm

Lemma 2.1Suppose that assumption () holds. Then the embedding ofEinis compact, where, is positive for a.e. .

Proof For any bounded set , there exists a positive constant such that for all . We claim that K is precompact in . In fact, since , for any , there exists such that

For any , applying the Hölder inequality for r such that and the first inequality in (5), we have

(6)

Besides, since is compactly embedded in , where , there are such that for any ,

(7)

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. . □

Lemma 2.2Assume that assumptions (), (), () and () hold andinE. Then

in.

Proof Assume that in E. Then, by Lemma 2.1,

in , where , is positive for a.e. . Passing to a subsequence if necessary, it can be assumed that

It is clear that

(8)

and

(9)

for all . Since is a Cauchy sequence in , so by (9) we know that is also a Cauchy sequence in . Therefore, by the completeness of , there exists such that in . Now we show that

(10)

for all and almost every . If not, there exist and , with , such that

for all . Then there exist a constant and , with , such that

for all . By the definition of , we have

for all and . Therefore, one has

Letting , we get

which contradicts the fact that for a.e. . Now we have proved (10). It follows from () that there exists such that

(11)

for all and . By (), there exists such that

(12)

for all and , which together with () shows there exists such that

(13)

for all and . Combining (11) and (13), we have

(14)

for all and . Hence, by (10) one has

for all and . It follows that

(15)

for all . By the Hölder inequality, we have

(16)

Similarly, we can prove

(17)

also

(18)

It follows from (15), (16), (17) and (18) that

which together with Lebesgue’s convergence theorem shows

(19)

as . Now we have proved the lemma. □

In the proof of Theorem 1.2, the following lemma is needed.

Lemma 2.3Assume thatis an open set. Then, for any closed set, there exists a functionsuch thatfor all, for allandfor all.

Proof Letting

then and . For any given , defining α and as follows,

one has , and . Denoting

and

it is clear that and . Lastly, we define

and

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.

Lemma 2.4Supposeand, then we have

as.

Proof It is clear that , so there exists such that

(20)

as . By the definition of , there exists with such that

(21)

Since is bounded, then there exists such that

as . Now, since is a basis of E, it follows that for all ,

as , which shows that . By Lemma 2.1 we have

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[19]

LetEbe a real Banach space andsatisfying thecondition. If Φ is bounded from below,

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.

Definition 2.6 Let and . The function Φ satisfies the condition if any sequence , such that

contains a subsequence converging to a critical point of Φ.

Now we show the following dual fountain theorem.

Lemma 2.7[20]

Ifand for every, there existssuch that

(i) ,

(ii) ,

(iii) as.

Moreover, ifsatisfies thecondition for all, then Φ has a sequence of critical pointssuch thatas.

### 3 Proof of theorems

Define the functional by

(22)

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 [1], by which we are led to study a variable functional that does not present such a strong indefinite nature.

Now we recall this method. For any , consider the linear functional defined as

By the Hölder inequality and using the second inequality in (5), we have

So, is continuous on . Set

for all . Obviously, is bilinear, bounded and coercive. Hence, the Lax-Milgram theorem implies that for every , there exists a unique such that

for any , that is,

for any . Using integration by parts, we get

for any , therefore

(23)

in a weak sense. We can write an integral expression for in the form

for any (see [21], Theorem 1); by density it can be extended for any (see Lemma 2.1 of [22]). Clearly, and for all .

It follows from (23) that

(24)

and by the Hölder inequality, we have

and it follows that

(25)

Hence,

(26)

So, we can consider the functional defined by . By (24), the reduced functional takes the form

(27)

By (12), we have

(28)

for all and , where and . Let , then , the space of continuous function u on , such that as . Therefore there exists such that

(29)

for all . Hence, one has

which together with (26) shows that Φ is well defined. Furthermore, it is well known that Φ is a functional with derivative given by

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, [1].

Lemma 3.1Under conditions (), (), (), (), Φ satisfies thecondition.

Proof Assume that is a sequence such that

Then there exists such that

for all .

Firstly, we show that is bounded. By (14), we have

(30)

for all and , which together with implies

(31)

Noting that for all , so is bounded.

By the fact that is bounded in E, there exists and a constant such that

(32)

and

in E as . It is obvious that

(33)

and

(34)

as . On the other hand, by (), (32) and Lemma 2.2, one has

(35)

as , which implies

(36)

as . Summing up (33) and (36), we have

(37)

as . By the Hölder inequality and (25), one gets

Then by Lebesgue’s convergence theorem, we have

as , which together with (34) implies

(38)

as . By Lemma 2.2 and (32), we get

as . Moreover, an easy computation shows that

Consequently, as . Φ satisfies the condition. □

Remark 3.2 Under conditions (), (), (), (), Φ satisfies the condition. Assume that is a sequence such that is bounded and

as . Then there exists such that

for all . The rest of the proof is the same as that of Lemma 3.1.

Proof of Theorem 1.2 For any , we take k disjoint open sets such that

For any and , there exist a closed set and an open set such that and

For every (), by Lemma 2.3 there exists such that and . Letting , can be extended to be a basis . Therefore , where . Now we define , .

By Lemma 3.1, satisfies the condition and . Hence, to prove Theorem 1.2, we should just show that Φ has the geometric property (i), (ii) and (iii) in Lemma 2.7.

(i) By Lemma 2.4

as for and . In view of (30) and the fact that , we have

(39)

Let , , , then as . Hence, we have

(40)

when and . Now we can choose , then as . When k is large enough, we have , , which together with (40) shows

(ii) For any , there exists such that

Then we have

(41)

and also

(42)

Since all the norms of a finite dimensional space are equivalent, there is a constant such that

for all . By (30), one has

Therefore, we have

(43)

For any with , we can choose small enough such that for all and , which together with () implies

(44)

for all and . Combining (24), (41), (42), (43) and (44), we have

for all with , when ε and are both small enough. Since , we can choose small enough such that

(iii) By (40), for any with , we have

Therefore

Since as , we have

as .

Hence, by Lemma 2.7, we obtain that problem (1) has infinitely many solutions satisfying

as . □

Proof of Theorem 1.5 Similar to (31), there exist constants , , such that

(45)

for all . Since , it follows from (45) that the functional Φ is bounded from below. By Lemma 2.5 and Remark 3.2, Φ possesses a critical point u satisfying

It remains to show that u is nontrivial. For every , there exist an open set G and a closed set H such that and

By Lemma 2.3, there exists a function such that and , , then . Choosing , then for all , which together with (28) shows

for all . Therefore, one has

(46)

In view of , we have for all , which together with () implies

(47)

for all . It follows from (24), (46), (47) that

when ε and λ are both small enough. Since , then . Hence, is a nontrivial solution of problem (1). □

### Competing interests

The author declares that she has no competing interests.

### Acknowledgements

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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