# Sub-super solutions for (p-q) Laplacian systems

Somayeh Haghaieghi1* and Ghasem Alizadeh Afrouzi2

Author Affiliations

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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

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Boundary Value Problems 2011, 2011:52  doi:10.1186/1687-2770-2011-52

 Received: 13 August 2011 Accepted: 2 December 2011 Published: 2 December 2011

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

In this work, we consider the system:

where Ω is a bounded region in RN with smooth boundary ∂Ω, Δp is the p-Laplacian operator defined by Δpu = div (|∇u|p-2u), p, q > 1 and g (x) is a C1 sign-changing the weight function, that maybe negative near the boundary. f, h, a, b are C1 non-decreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0. Using the method of sub-super solutions, we prove the existence of weak solution.

### 1 Content

In this paper, we study the existence of positive weak solution for the following system:

(1)

where Ω is a bounded region in RN with smooth boundary ∂Ω, Δp is the p-Laplacian operator defined by Δpu = div(|∇u|p-2 u), p, q > 1 and g(x) is a C1 sign-changing the weight function, that maybe negative near the boundary. f, h, a, b are C1 non-decreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0.

This paper is motivated by results in [1-5]. We shall show the system (1) with sign-changing weight functions has at least one solution.

### 2 Preliminaries

(Al) as s → ∞, ∀M > 0

(A2) lim f (s) = lim h (s) = ∞ as s → ∞.

(A3) as s → ∞.

Let λp, λq be the first eigenvalue of -Δp, -Δq with Dirichlet boundary conditions and φp, φq be the corresponding positive eigenfunctions with ||φp||= ||φq||= 1.

Let m, δ, γ, μp, μq > 0 be such that

(2)

and

(3)

We assume that the weight function g(x) take negative values in Ωδ, but it requires to be strictly positive in Ω-Ωδ. To be precise, we assume that there exist positive constants β and η such that g(x) ≥-β on and g(x) ≥ η on Ω-Ωδ. Let s0 ≥ 0 such that ηa(s) + f (s) > 0, ηb(s) + h(s) > 0 for s > s0 and

For γ such that γr-1 t > s0; t = min {αp, αq}, r = min{p, q} we define

where and .

We use the following lemma to prove our main results.

Lemma 1.1 [6]. Suppose there exist sub and supersolutions (ψ1, ψ2) and (z1, z2) respectively of (1) such that (ψ1, ψ2) ≤ (z1, z2). then (1) has a solution (u, v) such that (u, v) ∈ [(ψ1, ψ2), (z1, z2)].

### 3 Main result

Theorem 3.1Suppose that (A1)-(A3) hold, then for every λ ∈ [A, B], system (1) has at least one positive solution.

Proof of Theorem 3.1 We shall verify that (ψ1, ψ2) is a sub solution of (1.1) where

Let with w ≥ 0. Then

(4)

Now, on by (2),(3) we have

Since λ B then

thus

then by (4)

A similar argument shows that

Next, on . Since λ A, then

so we have

Then by (4) on we have

A similar argument shows that

We suppose that κp and κq be solutions of

respectively, and μ'p = ||κp||κ, ||κq||κ = μ'q.

Let

Let with w ≥ 0.

For sufficient C large

then

Similarly, choosing C large so that

then

Hence by Lemma (1.1), there exist a positive solution (u, v) of (1) such that (ψ1, ψ2) ≤ (u, v) ≤ (z1, z2).

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

SH has presented the main purpose of the article and has used GAA contribution due to reaching to conclusions. All authors read and approved the final manuscript.

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