### 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:

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
*f*; see, for example, [3-11]. Since (1) is set on

Moreover, in [12] the authors considered system (1) with periodic potential

There are also some papers treating the case with nonradial potential
*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
*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*
*where*
*is a constant*. *For every*

(
*where*
*is a positive function such that*
*and*

*Then problem* (1) *has infinitely many nontrivial solutions*
*satisfying*

*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 1.2***Assume that**V**satisfies*

(
*and*

*and**f**satisfies the following conditions*.

(
*There exist constants*
*and a function*
*such that*

*for all*
*and*

(
*There exist constants*
*and a function*
*such that*

*for all*
*and*

(
*For every*
*there exist a constant*
*and a function*
*such that*

*for all*
*and*

(
*There exist constants*
*and*
*such that*

*for all*
*and*
*where*

(
*for all*
*and*

*Then problem* (1) *has infinitely many nontrivial solutions*
*satisfying*

*as*

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*

*Then problem* (1) *has infinitely many nontrivial solutions*
*satisfying*

*as*

**Remark 1.4** In Theorem 1.2, infinitely many solutions for problem (1) are obtained under the
symmetry condition (
*a* is assumed to be positive, while in (
*a* is indefinite.

Moreover, under all the conditions of Theorem 1.2 except (

**Theorem 1.5***Assume that**L**satisfies* (
*and**W**satisfies* (
*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
*V* such as (
*V* and *f* which satisfy Theorem 1.5, but do not satisfy the corresponding results in [2-16]. For example,

and

in which
*π* the area of the unit ball in

which means that

### 2 Preliminary results

In order to establish our results via critical point theory, we firstly describe some
properties of the 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

for all
*R* under the norm

Let

where

**Lemma 2.1***Suppose that assumption* (
*holds*. *Then the embedding of**E**in*
*is compact*, *where*
*is positive for a*.*e*.

*Proof* For any bounded set
*K* is precompact in

For any
*r* such that

Besides, since

Now it follows from (6) and (7) that *K* is precompact in
*E* is compact embedded in

**Lemma 2.2***Assume that assumptions* (
*and* (
*hold and*
*in**E*. *Then*

*in*

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

in

It is clear that

and

for all

for all

for all

for all

for all

Letting

which contradicts the fact that

for all

for all

for all

for all

for all

for all

Similarly, we can prove

also

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

which together with Lebesgue’s convergence theorem shows

as

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*

*Proof* Letting

then
*α* and

one has

and

it is clear that

and

then

Since *E* is a Hilbert space, then there exists a basis

**Lemma 2.4***Suppose*
*and*
*then we have*

*as*

*Proof* It is clear that

as

Since

as
*E*, it follows that for all

as

in

We obtain the existence of a solution for problem (1) by using the following standard minimizing argument.

**Lemma 2.5**[19]

*Let**E**be a real Banach space and*
*satisfying the*
*condition*. *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

**Definition 2.6** Let

contains a subsequence converging to a critical point of Φ.

Now we show the following dual fountain theorem.

**Lemma 2.7**[20]

*If*
*and for every*
*there exists*
*such that*

(i)

(ii)

(iii)
*as*

*Moreover*, *if*
*satisfies the*
*condition for all*
*then* Φ *has a sequence of critical points*
*such that*
*as*

### 3 Proof of theorems

Define the functional

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

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

So,

for all

for any

for any

for any

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

for any

It follows from (23) that

and by the Hölder inequality, we have

and it follows that

Hence,

So, we can consider the functional

By (12), we have

for all
*u* on

for all

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

It can be proved that

**Lemma 3.1***Under conditions* (
*satisfies the*
*condition*.

*Proof* Assume that

Then there exists

for all

Firstly, we show that

for all

Noting that

By the fact that
*E*, there exists

and

in *E* as

and

as

as

as

as

Then by Lebesgue’s convergence theorem, we have

as

as

as

Consequently,

**Remark 3.2** Under conditions (

as

for all

*Proof of Theorem 1.2* For any
*k* disjoint open sets

For any

For every

By Lemma 3.1,

(i) By Lemma 2.4

as

Let

when
*k* is large enough, we have

(ii) For any

Then we have

and also

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

for all

Therefore, we have

For any