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Using the method of sub-super solutions, we prove the existence of weak solution.. We assume that the weight function gx take negative values inΩδ, but it requires to be strictly positiv

RESEARCH Open Access

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

Somayeh Haghaiegh1*and Ghasem Alizadeh Afrouzi2

* Correspondence:

Haghaieghi_ch86@yahoo.com

1 Department of Mathematics,

Science and Research Branch,

Islamic Azad University, Tehran, Iran

Full list of author information is

available at the end of the article

Abstract

In this work, we consider the system:

⎧

⎨

⎩

− p u = λ[g(x)a(u) + f (v)] in 

− q v = λ[g(x)b(v) + h(u)] in 

whereΩ is a bounded region in RN

with smooth boundary∂Ω, Δpis the p-Laplacian operator defined byΔpu = 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 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:

⎧

⎨

⎩

− p u = λ[g(x)a(u) + f (v)] in 

− q v = λ[g(x)b(v) + h(u)] in 

(1)

whereΩ is a bounded region in RN

with smooth boundary∂Ω, Δpis the p-Laplacian operator defined byΔpu = 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-decreas-ing functions satisfynon-decreas-ing 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) lim

f

⎛

⎝M(h(s)) q−11

⎞

⎠

s p−1 = 0

as s® ∞, ∀M > 0 (A2) lim f (s) = lim h (s) = ∞ as s ® ∞

(A3) lims a(s) p−1 = lim b(s) s q−1 = 0 as s ® ∞

Let lp, lqbe the first eigenvalue of -Δp, -Δq with Dirichlet boundary conditions and

p,qbe the corresponding positive eigenfunctions with ||p||∞= ||q||∞= 1

© 2011 Haghaiegh 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

Let m, δ, g, μp,μq > 0 be such that

|∇ϕ p|p − λ p ϕ p ≥ m in  δ

and

|∇ϕ q|q − λ q ϕ q ≥ m in δ

 δ ={x ∈ ; d(x, ∂) ≤ δ}.

We assume that the weight function g(x) take negative values inΩδ, but it requires

to be strictly positive inΩ-Ωδ To be precise, we assume that there exist positive

con-stants b and h such that g(x) ≥-b on  δ and g(x) ≥ h on Ω-Ωδ Let s0 ≥ 0 such that

ha(s) + f (s) > 0, hb(s) + h(s) > 0 for s >s0 and

f0= max{0, −f (0)}, h0= max{0, −h(0)}.

For g such that gr-1t > s0; t = min {ap, aq}, r = min{p, q} we define

A = max

⎡

⎢

⎢

⎢

⎢

⎣

γ λp ηa

⎛

⎜

⎜

⎝γ

1

p− 1αp

⎞

⎟

⎟

⎠+f

⎛

⎜

⎜

⎝γ

1

q− 1αq

⎞

⎟

⎟

⎠

ηb

⎛

⎜

⎜

⎝γ

1

q− 1αq

⎞

⎟

⎟

⎠+h

⎛

⎜

⎜

⎝γ

1

p− 1αp

⎞

⎟

⎟

⎠

⎤

⎥

⎥

⎥

⎥

⎦

B = min

⎡

⎢

⎢

⎢

⎣

mγ βa

⎛

⎜

⎝γ

1

p− 1

⎞

⎟

⎠+f0

βb

⎛

⎜

⎝γ

1

q− 1

⎞

⎟

⎠+h0

⎤

⎥

⎥

⎥

⎦

where α p= p−1p μ p

p

p−1 and α q= q−1q μ q q−1q .

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) Î [(ψ1,ψ2), (z1, z2)]

3 Main result

Theorem 3.1Suppose that (A1)-(A3) hold, then for every l Î [A, B], system (1) has at

least one positive solution

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

ψ1 =γ p −1 p−11

p ϕ p

p

p−1

ψ2 =γ q −1 q−11 ϕ q

q

q−1.

Let W ∈ H0 () with w ≥ 0 Then



 |∇ψ1|p−2∇ψ1∇wdx = γ



Now, on  δ by (2),(3) we have

γ (λ p ϕ p − |∇ϕ p|p)≤ −mγ

Since l ≤ B then

βa(γ p−11 ) + f0

thus

γ (λ p ϕ p − |∇ϕ p|p)≤ −mγ

≤ λ



−βa (γ p−11 )− f0



≤ λ



g(x)a (γ p−11 )− f0



λ



g(x) a



p−1

p γ p−11 ϕ p

1

p−1



+ f



q−1

q γ q−11 ϕ q

1

q−1



then by (4)



δ |∇ψ1|p−2∇ψ1∇wdx ≤



δ λ



g(x) a



p−1

p γ p−11 ϕ p

p

p−1



+ f



q−1

q γ q−11 ϕ q

q

q−1



wdx

A similar argument shows that



δ |∇ψ2|q−2∇ψ2∇wdx ≤



δ λ



g(x) b



q−1

q γ q−11 ϕ q

1

q−1



+ h



p−1

p γ p−11 ϕ p

1

p−1



wdx

Next, on  −  δ Since l≥ A, then

ηa



γ p−11 α p



+ f



γ q−11 α q



so we have

γ (λ p ϕ p − |∇ϕ p|p) ≤ γ λ p

≤ λ



ηa



γ p−11 α p



+ f



γ q−11 α q



≤ λ[g(x)a(ψ1) + f ( ψ2)],  −  δ

Then by (4) on we have

− p ψ1≤ λ[g(x)a(ψ1) + f ( ψ2)] on −  δ

A similar argument shows that

− q ψ2≤ λ[g(x)b(ψ2) + h( ψ1)]

We suppose thatpand qbe solutions of

− p κ p= 1 in

κ p= 0 on∂

− q κ q= 1 in

κ q= 0 on∂

respectively, and μ’p= ||p||, ||q||=μ’q Let

(z1, z2) =

⎛

⎝ c

μ

p

λ p−11 κ p,



2h



c λ q−11

 1

q−1

λ q−11 κ q

⎞

⎠

Let W ∈ H0 () with w ≥ 0

For sufficient C large

μ

p p−1

⎡

⎣||g||∞a



Cλ p−11



+ f



2h(C λ p−11

 1

q−1

λ q−11 μ

q

⎤

⎦

then



|∇z1| p−2 ∇z1∇wdx = λ



C

μp

p−1

wdx

≥ λ

⎡

⎣||g||∞ a (C λ p−11 ) + f

⎛

⎝(2h (Cλ p−11 ))

1

q−1

λ q−11 μ

q

⎞

⎠

⎤

⎦ dx

≥ λ

 ⎡

⎣g(x) a (Cλ p−1 κ1 p

μp ) + f

⎛

⎝(2h (Cλ p−11 ))

1

q−1

λ q−11 κ q

⎞

⎠

⎤

⎦ dx

=



[g(x) a (z1) + f (z2)] wdx

Similarly, choosing C large so that

||g||∞

⎛

⎝b2h



C λ p−11

 1

q−1

λ q−11 μ

q

⎞

⎠

h



C λ

1

p−1



|∇z2|q−2∇z2∇wdx = 2λh



C λ p−11  

wdx

≥ λ 

||g||∞b(z2) + h(z1)

wdx.

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

≤ (u, v) ≤ (z1, z2)

Author details

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

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.

Competing interests

The authors declare that they have no competing interests.

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

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doi:10.1186/1687-2770-2011-52 Cite this article as: Haghaiegh and Afrouzi: Sub-super solutions for (p-q) Laplacian systems Boundary Value Problems 2011 2011:52.

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