#### Abstract

We deal with the existence of constant sign solutions for the following variable exponent system Neumann boundary value problem: in in on . We give several sufficient conditions for the existence of the constant sign solutions, when satisfies neither sub-() growth condition, nor Ambrosetti-Rabinowitz condition (subcritical). In particular, we obtain the existence of eight constant sign solutions.

#### 1. Introduction

In recent years, there is a lot of interest in the study of various mathematical problems with variable exponent (see [1â€“32]). We refer readers to [1, 23, 31, 33] for the background of these problems. On the existence of solutions for elliptic systems with variable exponent, we refer to [15, 25, 27]. In this paper, we consider the existence of constant sign solutions for the following problem. where is an open bounded domain and possesses the cone property, and , is called the -Laplacian, is a positive parameter, and is the outward unit normal to . satisfies

We make the following assumption.

(A_{0}) is a nontrivial nonnegative singular coefficient; and satisfies
where , , on , , , and satisfies
where the notation means the conjugate function of , namely, , and

When (a constant), -Laplacian is the usual -Laplacian. The -Laplacian is nonhomogeneity. Because of its nonhomogeneity, the -Laplacian possesses more complicated nonlinearity than the -Laplacian. Many results and methods for -Laplacian problems do not hold for -Laplacian problems anymore. For the following examples.

() if is an open bounded domain, the Rayleigh quotient is zero in general. Only under some special conditions, we have . For example, if and only if is monotone in one-dimensional case (i.e., ) (see [11]). It is well known that the fact that is very important in the study of -Laplacian problems. For instance, in [34], the first eigenvalue and first eigenfunction are used to discuss the existence of positive solutions of -Laplacian problems successfully. But the -Laplacian does not have the first eigenvalue and first eigenfunction in general.

() The norm in is of Luxemburg type (we will explain later in Section 2). It is easy to see that for some . Hence the integral and the norm cannot keep the constant exponent relationship. It implies that we will have more difficulties in the study of -Laplacian problems. For example, it is very difficult to get the best Sobolev imbedding constant when we deal with the critical Sobolev exponent problems. Even if the best Sobolev imbedding constant could be obtained, it is also very hard to be applied to study the critical exponent problems.

() In [35], the authors applied the homogeneous transformation method to discuss the existence of positive solutions for a class of superlinear semipositon systems. Nonetheless, the -Laplacian is nonhomogeneity; this method is very hard to be used on the -Laplacian problems.

On the existence of constant sign solutions of -Laplacian problems, we refer to [35â€“40]. On the results of the constant sign solutions of variable exponent differential equations, we refer to [14, 19, 29].

Regarding the existence of solutions of , if satisfies the sub-() growth condition, that is, then the corresponding functional of is coercive; if satisfies the super-() growth condition (subcritical), that is, the following Ambrosetti-Rabinowitz condition: where positive constants and satisfy then the corresponding functional of satisfies Palais-Smale conditions (see [15, 25]). If satisfies the subcritical growth condition, but it satisfies neither the sub-() growth condition nor the super-() growth condition, then it would be difficult to testify that the corresponding functional is coercive or satisfying Palais-Smale conditions; the results in this case are rare.

In this paper, we deal with the existence of constant sign solutions of the problem , when the corresponding functional neither is coercive nor satisfies Ambrosetti-Rabinowitz condition. For example, we discuss the existence of solutions of , when satisfies sub-() growth condition near the origin in local; that is, the following condition or satisfies super-() growth condition in local (subcritical growth); that is, the following condition: where positive functions and satisfy

In particular, we get the existence of eight constant sign solutions of .

This paper is divided into four sections. In Section 2, we introduce some basic properties of the variable exponent Lebesgue-Sobolev spaces. In Section 3, several properties of -Laplacian are presented. In Section 4, we give the existence results of constant sign solutions of problem .

#### 2. Preliminary Results and Notations

Throughout this paper, the letters , ,â€‰â€‰,â€‰â€‰, denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process.

In order to discuss the problem , we need some theories on space which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces and -Laplacian which we will use later (for details, see [6, 10, 12, 13]). Write We introduce the norm on by and () becomes a Banach space; we call it variable exponent Lebesgue space.

Proposition 1 (see [6]). *(i)â€‰â€‰The space is a separable, uniform convex Banach space, and its conjugate space is , where . For any and , one has
**(ii)â€‰â€‰If ,â€‰â€‰, for any , then , and the imbedding is continuous.*

Denote with the norm where , ; then is a Banach space.

Proposition 2. *If is a Caratheodory function, that is, satisfies the following:*(i)*for a.e. *, * is a continuous function from *â€‰* to*â€‰â€‰;(ii)*for any *, * is measurable.**If there exist , , and positive constant such that
**
then the Nemytsky operator from to defined by is a continuous and bounded operator.*

*Proof. *Similar to the proof of [41], we omit it here.

Proposition 3 (see [6]). *If one denotes
**
then*(i);
(ii);
;
(iii);
.

Proposition 4 (see [6]). *If , , , then the following statements are equivalent to each other:*(1)*;*(2)*;*(3) * in measure in ** and **.*

The spaces and are defined by and endowed with the following norm:

We denote by the closure of in .

Proposition 5 (see [6]). * is a separable reflexive Banach space.**(ii)â€‰â€‰If and for any , then the imbedding from to is compact and continuous.*

Let , , and for a.e.â€‰â€‰. Define with the norm then is a Banach space.

Proposition 6 (see [12]). *Assume that the boundary of possesses the cone property and . Suppose that , for a.e. , , and . If and
**
then the embedding is compact.*

Denote . The norm on is defined by

For any and in , let Then where

The integral functional associated with the problem is

Without loss of generality, we may assume that , . Obviously, We have
where denotes the partial derivative of with respect to its th variable; then the condition (A_{0}) holds

From Propositions 2 and 6 and condition (A_{0}), it is easy to see that and satisfies
where

We say is a critical point of if

The dual space of will be denoted by ; then for any , there exists , such that . We denote , , and to be the norms of , and , respectively. It is well known, and . Therefore

Proposition 7. *(i)â€‰â€‰If satisfies
**
then .**(ii)â€‰â€‰If satisfies
**
then .*

*Proof. *(i) Similar to the proof of [15], we omit it here.

#### 3. Properties of Operators

In this section, we will discuss the properties of -Laplacian and Nemytsky operator.

From Propositions 2 and 6, we can easily see that .

Proposition 8 (see [25]). *(i)â€‰â€‰ is a convex functional;**(ii)â€‰â€‰ is strictly monotone; that is, for any , with , we have
**(iii)â€‰â€‰ is a mapping of type (); that is, if in and
**
then in ;**(iv)â€‰â€‰ is a bounded homeomorphism.*

Theorem 9. *If the assumption (A _{0}) is satisfied, then . Moreover, and are weakly-strongly continuous; that is, in implies and in .*

*Proof. *Suppose is a weak convergent sequence in . From Proposition 6, we can conclude that is strong convergent in , where and satisfies

From Proposition 2, we can see that is weakly-strongly continuous, and in .

Since is a separable and reflexive Banach space, there are and such that

For convenience, we write

*Definition 10. *One says that satisfies (PS) condition in , if any sequence such that is bounded and , as , has a convergent subsequence.

One assumes satisfies the following condition:(B)there exist functions satisfying ,â€‰â€‰ on and

Lemma 11. *If () is large enough, (A _{0}) and (B) are satisfied; then satisfies (PS) condition on .*

*Proof. *It follows from (B) that

Denote

It is easy to see that .

Let be a (PS) sequence. By computation,

From Young inequality, we have

Similarly, we have

Assume is a small enough positive constant and is large enough; we have

Thus and are bounded. Therefore has a weak convergent subsequence (which we still denote by ) such that as . According to Theorem 9, we have as . Since as , we have as . Since is a homeomorphism, we have that is strong convergent in .

#### 4. Existence and Multiplicity of Solutions

In this section, using the critical point theory, we will discuss the existence and multiple existence of constant sign solutions of problem .

*Definition 12. *One calls is a weak solution of if

It is easy to see that the critical point of is a solution of .

Denote , , where . For any , we say belong to the first, the second, the third, or the fourth quadrant of , if and , and , and , and and , respectively.

*Definition 13. *(i) One calls that satisfies sub-() growth condition near the origin in the first quadrant of , if it satisfies the following.

(A_{1})
where , on .

(ii) One calls that satisfies super-() growth condition near the infinity in the first quadrant of , if it satisfies

(A_{2}) for and with .

*
Remark.* (i) Similarly, we can give the definitions of satisfying sub-() growth condition near the origin, super-() growth condition near the origin or near the infinity in the second, the third, and the fourth quadrant of , respectively.

(ii) We say satisfies sub-() growth condition near the origin, super-() growth condition near the infinity in , if satisfies corresponding growth condition in every quadrant of .

(iii) We say satisfies some growth condition in local, if it satisfies some growth condition in a quadrant.

We will discuss the existence of solutions in the following three cases:â€‰Case (I): satisfies sub-() growth condition near the origin in local;â€‰Case (II): satisfies super-() growth condition near the infinity in local;â€‰Case (III): satisfies sub-() growth condition near the origin and super-() growth condition near the infinity in local.

##### 4.1. Case (I)

We assume(S) satisfies , , with .

Theorem 14. *If is large enough and satisfies (A _{0}), (S), and sub-() growth condition near the origin in the first quadrant of , then the problem has a nontrivial constant sign solution in the first quadrant of .*

*Proof. *It is easy to check that , and

Let us consider the following auxiliary problem:

The corresponding functional is
where

We divide into several disjoint subsets which satisfy

In the following, we denote

Suppose is small enough. By (A_{0}) and Proposition 6, we have

Note that and . When is small enough, it follows from Proposition 3 that

Obviously,