# Existence of positive solutions for variable exponent elliptic systems

Author Affiliations

1 Department of Mathematics, Sciences and Research, Islamic Azad University (IAU) Tehran, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

3 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

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Boundary Value Problems 2012, 2012:37  doi:10.1186/1687-2770-2012-37

 Received: 30 October 2011 Accepted: 3 April 2012 Published: 3 April 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We consider the system of differential equations

where Ω ⊂ ℝN is a bounded domain with C2 boundary ∂Ω, 1 < p(x) ∈C1 is a function. is called p(x)-Laplacian. We discuss the existence of positive solution via sub-super solutions without assuming sign conditions on f(0), h(0).

MSC: 35J60; 35B30; 35B40.

##### Keywords:
positive solutions; p(x)-Laplacian problems; sub-supersolution

### 1. Introduction

The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see[1-3]). Many results have been obtained on this kind of problems, for example [1,3-8]. In [7], Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to [8,9]. In this article, we mainly consider the existence of positive weak solutions for the system

where Ω ⊂ ℝN is a bounded domain with C2 boundary Ω, 1 < p(x) ∈ C1 is a function. The operator is called p(x)-Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the well-known p-Laplacian system. There are many articles on the existence of solutions for p-Laplacian elliptic systems, for example [5,10]. Owing to the nonhomogeneity of p(x)-Laplacian problems are more complicated than those of p-Laplacian, many results and methods for p-Laplacian are invalid for p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient

is zero in general, and only under some special conditions λp(x) > 0 (see [11]), and maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist, but the fact that the first eigenvalue λp > 0 and the existence of the first eigenfunction are very important in the study of p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.

Hai and Shivaji [10], consider the existence of positive weak solutions for the following p-Laplacian problems

the first eigenfunction is used to construct the subsolution of p-Laplacian problems success-fully. On the condition that λ is large enough and

the authors give the existence of positive solutions for problem (I).

Chen [5], considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:

the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following

(i) If α, β ≥ 0, γ, δ > 0, θ = (p - 1 - α)(q - 1 - β) - γδ > 0, then problem (II) has a positive weak solution for each λ > 0;

(ii) If θ = 0 and pγ = q(p - 1 - α), then there exists λ0 > 0 such that for 0 < λ < λ0, then problem (II) has no nontrivial nonnegative weak solution.

On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exist, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Zhang [12] investigated the existence of positive solutions of the system

In this article, we consider the existence of positive solutions of the system

where p(x) ∈ C1 is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝN is a bounded domain.

To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [6,13]). If Ω ⊂ ℝN is an open domain, write

Throughout the article, we will assume that:

(H1) Ω ⊂ ℝN is an open bounded domain with C2 boundary Ω.

(H2) p(x) ∈ C1 and 1 < p- ≤ p+.

(H3) a, b C1([0, )) are nonnegative, nondecreasing functions such that

(H4) f, h : [0, +) → R are C1, monotone functions, limu→+∞ f(u) = +, limu→+∞ h(u) = +, and

(H5) g : [0, +) (0, +) is a continuous function such that , and

Denote

We introduce the norm on Lp(x)(Ω) by

and (Lp(x)(Ω), |.|p(x)) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x)(Ω), |.|p(x)) is a separable, reflexive, and uniform convex Banach space (see [[6], Theorems 1.10 and 1.14]).

The space W1,p(x)(Ω) is defined by W1,p(x)(Ω) = {u Lp(x) : | u| Lp(x)}, and it is equipped with the norm

We denote by W01,p(x)(Ω) is the closure of in W1,p(x)(Ω). W1,p(x)(Ω) and are separable, reflexive, and uniform convex Banach space (see [[6], Theorem 2.1] We define

then is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [[14], Theorem 3.1]).

If is called a weak solution of (P) if it satisfies

Define as

where l(x, u) is continuous on , and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [15], we have the following lemma.

Lemma 1.1. (Comparison Principle). Let u, v W1,p(x)(Ω) satisfying Au - Av ≥ 0 in . If (i.e., u ≥ v on ∂Ω ), then u ≥ v a.e. in Ω.

Here and hereafter, we will use the notation d(x, Ω) to denote the distance of x ∈ Ω to the boundary of Ω.

Denote d(x) = d(x, Ω) and . Since Ω is C2 regularly, then there exists a constant δ ∈ (0, 1) such that , and |d(x)| ≡ 1.

Denote

Obviously, . Considering

(1)

we have the following result

Lemma 1.2. (see [16]). If positive parameter η is large enough and w is the unique solution of (1), then we have

(i) For any θ∈ (0, 1) there exists a positive constant C1 such that

(ii) There exists a positive constant C2 such that

### 2. Existence results

In the following, when there be no misunderstanding, we always use Ci to denote positive constants.

Theorem 2.1. On the conditions of (H1) - (H5), then (P) has a positive solution when λ is large enough.

Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ1, Φ2) and supersolution (z1, z2) of (P), such that Φ1 ≤ z1 and Φ2 ≤ z2. That is (Φ1, Φ2) and (z1, z2) satisfies

and

for all with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [16]), then (P) has a positive solution.

Step 1. We construct a subsolution of (P).

Let σ ∈ (0, δ) is small enough. Denote

It is easy to see that . Denote

By computation

From (H3) and (H4), there exists a positive constant M > 1 such that

Let , then

(2)

If k is sufficiently large, from (2), we have

(3)

Let -λζ = , then

from (3), then we have

(4)

Since , then there exists a positive constant C3 such that

If k is sufficiently large, let -λζ = , we have

then

Since ϕ (x) 0 and a, f are monotone, when λ is large enough, then we have

(5)

Obviously

(6)

Combining (4), (5), and (6), we can conclude that

(7)

Similarly

(8)

From (7) and (8), we can see that (ϕ1, ϕ2) = (ϕ, ϕ) is a subsolution of (P).

Step 2. We construct a supersolution of (P).

We consider

where . We shall prove that (z1, z2) is a supersolution for (p).

For with q ≥ 0, it is easy to see that

(9)

Since ,when μ is sufficiently large, combining Lemma 1.2 and (H3), then we have

(10)

Hence

(11)

Also

By (H3), (H4), when μ is sufficiently large, combining Lemma 1.2 and (H3), we have

Then

(12)

According to (11) and (12), we can conclude that (z1, z2) is a supersolution for (P).

It only remains to prove that ϕ1 ≤ z1 and ϕ2 ≤ z2.

In the definition of v1(x), let . We claim that

(13)

From the definition of v1, it is easy to see that

and

It only remains to prove that

Since then there exists a point such that

If v1(x0) - ϕ(x0) < 0, it is easy to see that 0 < d(x0) < δ, and then

From the definition of v1, we have

It is a contradiction to ∇v1(x0) - ϕ(x0) = 0. Thus (13) is valid.

Obviously, there exists a positive constant C3 such that

Since , according to the proof of Lemma 1.2, then there exists a positive constant C4 such that

When is large enough, we have

According to the comparison principle, we have

(14)

From (13) and (14), when and λ ≥ 1 is sufficiently large, we have

(15)

According to the comparison principle, when μ is large enough, we have

Combining the definition of v1(x) and (15), it is easy to see that

When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that is large enough, then is large enough. Similarly, we have ϕ2 ≤ z2. This completes the proof. □

### 3. Asymptotic behavior of positive solutions

In this section, when parameter λ → +, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.

Theorem 3.1. On the conditions of (H1)-(H5), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then

(i) There exist positive constants C1 and C2 such that

(16)

(17)

(ii) for any θ ∈ (0, 1), there exist positive constants C3 and C4 such that

(18)

(19)

where μ satisfies (10).

Proof. (i) Obviously, when 2δ ≤ d(x), we have

then there exists a positive constant C1 such that

It is easy to see

then

Similarly

Thus (16) and (17) are valid.

(ii) Denote

where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.

Obviously, v3(x) ∈ C1ρ), By computation

where

Let where ρ > 0 is small enough, it is easy to see that

where ρ > 0 is small enough, then we have

Obviously v3(x) ≥ z1(x) on Ωρ. According to the comparison principle, we have v3(x) ≥ z1 (x) on Ωρ. Thus

Let when ρ > 0 is small enough, it is easy to see that

Similarly, when ρ > 0 is small enough, we have

Obviously, when d(x) < σ, we have

Thus (18) and (19) are valid. This completes the proof. □

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

The authors would like to appreciate the referees for their helpful comments and suggestions. The third author partly supported by the National Science Foundation of China (10701066 & 10971087).

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