### Abstract

This paper explores the existence of solutions for a class of

**MSC: **
34B10.

##### Keywords:

### 1 Introduction

In this paper, we consider the existence of solutions for the following system:

where

The study of differential equations and variational problems with variable exponent
growth conditions has attracted more and more attention in recent years. Many results
have been obtained on these problems, for example, [1-16]. We refer to [3,12,16] for the applied background of these problems. If
*p*-Laplacian. If

In recent years, because of the wide mathematical and physical background (see [17-19]), the existence of positive solutions for the *p*-Laplacian equation group has received extensive attention. Especially, when

Let

(i) For almost every

(ii) For each
*J*;

(iii) For each

Throughout the paper, we denote

The inner product in
*C*,
*W* will be equipped with the norm

We say a function
*P*) if
*P*) a.e. on *J* and the boundary value conditions.

In this paper, we always use

We say

where

We will discuss the existence of solutions for (*P*) in the following two cases:

(i)

(ii)

This paper is organized as follows. In Section 2, we do some preparation. In Section
3, we discuss the existence of solutions of (*P*). Finally, in Section 4, we discuss the existence of nonnegative solutions for (*P*).

### 2 Preliminary

For any

**Lemma 2.1** (see [5])

*is a continuous function and satisfies the following*:

(i) *For any*
*is strictly monotone*, *that is*,

(ii) *There exists a function*
*as*
*such that*

It is well known that

It is clear that

Let us now consider the following problem:

with the boundary value condition

where
*u* is a solution of (1) with (2), by integrating (1) from 0 to *t*, we find that

Denote

From (3), we have

By integrating (4) from 0 to *t*, we find that

From (2), we have

and

For fixed

It is easy to obtain the following lemma.

**Lemma 2.2**
*is continuous and sends bounded sets of*
*to bounded sets of*
*Moreover*,

It is clear that

Let us now consider another problem

with the boundary value condition

where

and

where

From

From

From (9) and (10), we have

For fixed

**Lemma 2.3***The function*
*has the following properties*:

(i) *For any fixed*
*the equation*

*has a unique solution*

(ii) *The function*
*defined in* (i), *is continuous and sends bounded sets to bounded sets*. *Moreover*,

*Proof* (i) It is easy to see that

From Lemma 2.1, it is immediate that

and hence, if (11) has a solution, then it is unique.

Let
*i*th component

Thus
*J* and

Obviously,

Thus the *i*th component

Let us consider the equation

It is easy to see that all the solutions of (12) belong to

which implies the existence of solutions of

In this way, we define a function

(ii) By the proof of (i), we also obtain that

It only remains to prove the continuity of
*C* and

Now, we define the operator

It is clear that

If *u* is a solution of (1) with (2), we have

and

If *u* is a solution of (7) with (8), we have

and

We denote

**Lemma 2.4***The operators*
*are continuous and send equi*-*integrable sets in*
*to relatively compact sets in*

*Proof* We only prove that the operator

It is easy to check that

it is easy to check that

Let now *U* be an equi-integrable set in

We want to show that

Let

Hence the sequence
*C*. According to the bounded continuous of the operator
*C*, then
*C*.

From the definition of
*C*. Thus,

Let us define

It is easy to see that

We denote

**Lemma 2.5**
*is a solution of* (*P*) *if and only if*
*is a solution of the following abstract equation*:

*Proof* If
*P*), according to the proof before Lemma 2.5, it is easy to obtain that
*S*).

Conversely, if
*S*), then

which implies

It follows from (*S*) that

then

By the condition of the mapping

and then

From (*S*), we have

and

then

Thus

From (*S*), we have

By the condition of the mapping

which implies that

Since

Moreover, from (*S*), it is easy to obtain

and

Hence
*P*).

This completes the proof. □

### 3 Existence of solutions

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions
for (*P*), when

We denote (*S*) as

where

**Theorem 3.1***If*
*satisfies a sub*-
*growth condition*, *then the problem* (*P*) *has at least one solution for any fixed parameter*

*Proof* Denote

where

According to Lemma 2.5, we know that (*P*) has the same solution of

when

It is easy to see that the operators
*W*.

We claim that all the solutions of (14) are uniformly bounded for
*n*

From Lemma 2.2, we have

which together with the sub-

From (14), we have

then

Denote

It follows from (14) and (15) that

For any

which implies that

Thus

It follows from (16) and (17) that

Similarly, we have

Thus,

Thus, we can choose a large enough

Denote

where

It is easy to see that

Obviously, (19) possesses a unique solution

Therefore (*P*) has at least one solution. This completes the proof. □

In the following, we investigate the existence of solutions for (*P*) when

Denote

Assume the following.

(A_{1}) Let a positive constant *ε* be such that

It is easy to see that
*W*. We have the following theorem.

**Theorem 3.2***Assume that* (A_{1}) *is satisfied*. *If positive parameters*
*and*
*are small enough*, *then the problem* (*P*) *has at least one solution on*

*Proof* Similarly, we denote

with (2) and (8) if and only if

From the proof of Theorem 3.1, we can see that
*W*. According to Leray-Schauder’s degree theory, we only need to prove that

(1^{∘})

(2^{∘})

then we can conclude that the system (*P*) has a solution on

(1^{∘}) If there exists a
*λ* satisfy

and

Since
*i* such that

(i) If

(

Note that
_{1}), we have

It is a contradiction to

(

Since

thus

From Lemma 2.2,

When
_{1}) and (21), we can conclude that

It is a contradiction. Thus

(ii) If

Summarizing this argument, for each

(2^{∘}) Since
_{1}) holds

This completes the proof. □

As applications of Theorem 3.2, we have the following.

**Corollary 3.3***Assume that*
*where*
*satisfy*
*If*
*and*
*are small enough*, *then the problem* (*P*) *possesses at least one solution*.

*Proof* It is easy to have

From

Since
*ε* such that

From Lemma 2.2 and the small enough

then

Similarly, we have

Obviously, it follows from

Thus, the conditions of (A_{1}) are satisfied, then the problem (*P*) possesses at least one solution. □

**Corollary 3.4***Assume that*
*where*
*satisfy*
*If*
*and*
*are small enough*, *then the problem* (*P*) *possesses at least one solution*.

*Proof* From Lemma 2.2, we have

Since

Similarly, we have

From

Thus, the conditions of (A_{1}) are satisfied, then the problem (*P*) possesses at least one solution. □

We denote

Assume the following.

(A_{2}) Let positive constants

It is easy to see that
*W*. We have the following.

**Corollary 3.5***Assume that*

*where**ϵ*,
*are positive constants*;
*satisfy*
*and*
*Then the problem* (*P*) *possesses at least one solution*.

*Proof* Similar to the proof of Theorem 3.2, we only need to prove that (A_{2}) is satisfied, then we can conclude that the problem (*P*) possesses at least one solution.

From

where we suppose

From Lemma 2.2, we have

then

From

Since

From Lemma 2.3, we have

then

Obviously, it follows from

Thus, the conditions of (A_{2}) are satisfied, then the problem (*P*) possesses at least one solution. □

**Corollary 3.6***Assume that*

*where**ϵ*,
*are positive constants*;
*satisfy*
*and*
*If*
*and*
*are small enough*, *then the problem* (*P*) *possesses at least one solution*.

*Proof* Similar to the proof of Corollary 3.5, we conclude that (A_{2}) is satisfied. Then the problem (*P*) possesses at least one solution. □

### 4 Existence of nonnegative solutions

In the following, we deal with the existence of nonnegative solutions of (*P*). For any