## Boundary Value Problems

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# Existence of positive solutions for variable exponent elliptic systems

Boundary Value Problems20122012:37

DOI: 10.1186/1687-2770-2012-37

Accepted: 3 April 2012

Published: 3 April 2012

## Abstract

We consider the system of differential equations

where Ω N is a bounded domain with C 2 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[13]). Many results have been obtained on this kind of problems, for example [1, 38]. 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 N 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 λ p (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 λ p > 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 N is a bounded domain.

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

Throughout the article, we will assume that:

(H1) Ω N 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, lim u →+∞ f(u) = +, lim u →+∞ h(u) = +, and

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

Denote
We introduce the norm on L p (x)(Ω) by

and (L p (x)(Ω), |.| p (x)) becomes a Banach space, we call it generalized Lebesgue space. The space (L p (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 L p (x): | u| L p (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 C 2 such that

## 2. Existence results

In the following, when there be no misunderstanding, we always use C i 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) C1 ρ ), 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 Ω ρ . According to the comparison principle, we have v3(x) ≥ z1 (x) on Ω ρ. 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. □

## Declarations

### 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).

## Authors’ Affiliations

(1)
Department of Mathematics, Sciences and Research, Islamic Azad University (IAU)
(2)
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran
(3)
Department of Mathematics and Information Science, Zhengzhou University of Light Industry
(4)
Department of Mathematics, Ferdowsi University of Mashhad

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