Open Access Research

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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Citation and License

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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/52


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

© 2011 Haghaieghi and Afrouzi; licensee Springer.

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M1">View MathML</a>

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M1">View MathML</a>

(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

In this article, we use the following hypotheses:

(Al) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M2">View MathML</a> as s → ∞, ∀M > 0

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

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M3">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M4">View MathML</a>

(2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M5">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M6">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M7">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M8">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M11">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M12">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M13">View MathML</a> with w ≥ 0. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M14">View MathML</a>

(4)

Now, on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M15">View MathML</a> by (2),(3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M16">View MathML</a>

Since λ B then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M17">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M18">View MathML</a>

then by (4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M19">View MathML</a>

A similar argument shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M20">View MathML</a>

Next, on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M21">View MathML</a>. Since λ A, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M22">View MathML</a>

so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M23">View MathML</a>

Then by (4) on we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M24">View MathML</a>

A similar argument shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M25">View MathML</a>

We suppose that κp and κq be solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M26">View MathML</a>

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

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M27">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M28">View MathML</a> with w ≥ 0.

For sufficient C large

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M29">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M30">View MathML</a>

Similarly, choosing C large so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M31">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/52/mathml/M32">View MathML</a>

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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