Abstract
In this paper I consider a class of sublinear SchrödingerMaxwell 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:
SchrödingerMaxwell equations; sublinear; minimizing theorem; dual fountain theorem1 Introduction and main result
Consider the following semilinear SchrödingerMaxwell equations:
Such a system, also known as the nonlinear SchrödingerPoisson 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 electrostatictype 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, [311]. 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 PalaisSmale 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
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
andfsatisfies the following conditions.
() There exist constants, and a functionsuch that
() There exist constants, and a functionsuch that
() For every, there exist a constantand a functionsuch that
() There exist constants, andsuch that
Then problem (1) has infinitely many nontrivial solutionssatisfying
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
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 [216]. For example,
and
in which . It is clear that is indefinite. Denoting by π the area of the unit ball in , we obtain
which means that . So, (2) satisfies our results, but does not satisfy the results in [317].
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
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
Besides, since is compactly embedded in , where , there are such that for any ,
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
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
and
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
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
which contradicts the fact that for a.e. . Now we have proved (10). It follows from () that there exists such that
for all and . By (), there exists such that
for all and , which together with () shows there exists such that
for all and . Combining (11) and (13), we have
for all and . Hence, by (10) one has
for all . By the Hölder inequality, we have
Similarly, we can prove
also
It follows from (15), (16), (17) and (18) that
which together with Lebesgue’s convergence theorem shows
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,
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
Proof It is clear that , so there exists such that
as . By the definition of , there exists with such that
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
Moreover, ifsatisfies thecondition for all, then Φ has a sequence of critical pointssuch thatas.
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 [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
for all . Obviously, is bilinear, bounded and coercive. Hence, the LaxMilgram theorem implies that for every , there exists a unique such that
for any . Using integration by parts, we get
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
and by the Hölder inequality, we have
and it follows that
Hence,
So, we can consider the functional defined by . By (24), the reduced functional takes the form
By (12), we have
for all and , where and . Let , then , the space of continuous function u on , such that as . Therefore there exists such that
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
Firstly, we show that is bounded. By (14), we have
for all and , which together with implies
Noting that for all , so is bounded.
By the fact that is bounded in E, there exists and a constant such that
and
and
as . On the other hand, by (), (32) and Lemma 2.2, one has
as . Summing up (33) and (36), we have
as . By the Hölder inequality and (25), one gets
Then by Lebesgue’s convergence theorem, we have
as , which together with (34) implies
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
Let , , , then as . Hence, we have
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
and also
Since all the norms of a finite dimensional space are equivalent, there is a constant such that
Therefore, we have
For any with , we can choose small enough such that for all and , which together with () implies
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
Hence, by Lemma 2.7, we obtain that problem (1) has infinitely many solutions satisfying
Proof of Theorem 1.5 Similar to (31), there exist constants , , such that
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
In view of , we have for all , which together with () implies
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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