Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient

Wu, Xianbin

doi:https://doi.org/10.1155/2014/396704

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 396704 | https://doi.org/10.1155/2014/396704

Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient

Xianbin Wu¹

Academic Editor: Leszek Gasinski

Received14 Nov 2013

Accepted22 Feb 2014

Published07 Apr 2014

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 [1–32]). 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.

(A₀) 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 [35–40]. 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 , , , , 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) The space is a separable, uniform convex Banach space, and its conjugate space is , where . For any and , one has
(ii) If , , 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 to ;(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) If and for any , then the imbedding from to is compact and continuous.

Let , , and for a.e. . 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 (A₀) holds

From Propositions 2 and 6 and condition (A₀), 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) If satisfies then .
(ii) 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) is a convex functional;
(ii) is strictly monotone; that is, for any , with , we have
(iii) is a mapping of type (); that is, if in and then in ;
(iv) is a bounded homeomorphism.

Theorem 9. If the assumption (A₀) 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 , on and

Lemma 11. If () is large enough, (A₀) 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.
(A₁) where , on .
(ii) One calls that satisfies super-() growth condition near the infinity in the first quadrant of , if it satisfies

(A₂) 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: Case (I): satisfies sub-() growth condition near the origin in local; Case (II): satisfies super-() growth condition near the infinity in local; 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 (A₀), (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 (A₀) and Proposition 6, we have
Note that and . When is small enough, it follows from Proposition 3 that
Obviously,
Noting that , without loss of generality, we may assume that .
Assume . Obviously
Assume . When is large enough, then we have
Thus, when is large enough, we have
Assume . We have
Assume . Similar to the above discussion, when is large enough, we have
Therefore, when is large enough, we have
It is easy to see that is weak lower semicontinuous. Therefore, can archive its infimum at some point . Obviously, . Taking as a test function for , it is easy to see that . Thus is nonnegative, and then it is a constant sign solution of .
Note that (50) is satisfied. Without loss of generality, we may assume that
Take which are nontrivial nonnegative. It is easy to see that
Thus has at least one nontrivial critical point in the first quadrant of with . Thus, is a nontrivial constant sign solution of . According to condition (S), it is easy to see that and are all nontrivial.

Theorem 15. If is large enough and satisfies (A₀), (S), and sub-() growth condition near the origin in , then problem has at least four nontrivial constant sign solutions.

Proof. (i) Similar to the proof of Theorem 14, we can see that has a nontrivial constant sign in the th quadrant of , such that , . According to condition (S), and are both nontrivial. Thus has at least four constant sign solutions.

4.2. Case (II)

Theorem 16. If is large enough and satisfies (A₀), (S), and the super-() growth condition near the infinity in the first quadrant of , then has a nontrivial constant sign solution in the first quadrant of .

Proof. It is easy to check that and
Let us consider the auxiliary problem .
The corresponding functional is where
We will prove that satisfies the conditions of Mountain Pass Lemma.
It is easy to see that satisfies for and , where
From Lemma 11, we can see that satisfies (PS) condition in .
From (29) and Proposition 7, we have
In the following, we denote
Without loss of generality, we may assume that
Similar to the proof of Theorem 14, when is large enough, we can get
For fixed with and , we have
Since and on , as . Obviously, ; then satisfies the conditions of Mountain Pass Lemma (see [42, 43]). So, we can conclude that has at least one nontrivial critical point with . Obviously, . Taking as a test function, it is easy to see that . Thus is nontrivial nonnegative; then it is a constant sign solution of . According to condition (S), it is easy to see that and are both nontrivial and satisfy .

Theorem 17. If is large enough and satisfies (A₀), (S), and the super-() growth condition near infinity in , then has four nontrivial constant sign solutions.

Proof. As satisfies the super-() growth condition in , then satisfies the super-() growth condition in every quadrant of . By Theorem 16, we can see that has a solution in the th quadrant of and satisfies and and () are both nontrivial.

4.3. Case (III)

Theorem 18. If is large enough and satisfies (A₀), (S), sub-() growth condition near the origin, and super-() growth condition near the infinity in the first quadrant of , then has two nontrivial constant sign solutions in the first quadrant of .

Proof. Similar to the proof of Theorem 16, we can get the existence of solution in the first quadrant in , which satisfies , and and both are nontrivial. By Theorem 14, has the second solution in the first quadrant in , which satisfies , and and both are nontrivial. Therefore, has at least two constant sign solutions.

Note. Let where and satisfy and are all changed sign functions, and then we can see that the functional satisfies (PS) condition and has a nontrivial constant sign solution, but does not satisfy the Ambrosetti-Rabinowitz condition, and it is not coercive.

Theorem 19. If is large enough and satisfies (A₀), (S), sub-() growth condition near the origin, and super-() growth condition near the infinity in , then has eight nontrivial constant sign solutions in .

Proof. Similar to the proof of Theorem 18, we can see that has nontrivial solutions and in the th quadrant in , which satisfy and , and and are all nontrivial. Thus has at least eight nontrivial constant sign solutions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partly supported by Natural Science Foundation of Ningbo (no. 2013A610276), the Professional Development Program of Zhejiang Province Visiting Scholar (no. FX2013117), National Natural Science Foundation of China (no. 11326161), and the key projects of Science and Technology Research of the Henan Education Department (no. 14A110011).

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Copyright © 2014 Xianbin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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