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

## Abstract

In this work, we consider the system:

${ - Δ p u = λ [ g ( x ) a ( u ) + f ( v ) ] in Ω - Δ q v = λ [ g ( x ) b ( v ) + h ( u ) ] in Ω u = v = 0 on ∂ Ω ,$

where Ω is a bounded region in RN with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = 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:

${ - Δ p u = λ [ g ( x ) a ( u ) + f ( v ) ] in Ω - Δ q v = λ [ g ( x ) b ( v ) + h ( u ) ] in Ω u = v = 0 on ∂ Ω ,$
(1)

where Ω is a bounded region in RN with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = 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.

This paper is motivated by results in [15]. We shall show the system (1) with sign-changing weight functions has at least one solution.

## 2 Preliminaries

(Al) $lim f M ( h ( s ) ) 1 q - 1 s p - 1 =0$ as s → ∞, M > 0

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

(A3) $lim a ( s ) s p - 1 =lim b ( s ) s q - 1 =0$ as s → ∞.

Let λ p , λ q be the first eigenvalue of -Δ p , -Δ q with Dirichlet boundary conditions and φ p , φ q be the corresponding positive eigenfunctions with ||φ p || = ||φ q || = 1.

Let m, δ, γ, μ p , μ q > 0 be such that

$| ∇ φ p | p - λ p φ p ≥ m in Ω ¯ δ φ p ≥ μ p on Ω - Ω δ$
(2)

and

${ | ∇ φ q | q - λ q φ q ≥ m in Ω ¯ δ φ p ≥ μ p on Ω - Ω δ .$
(3)
$Ω ¯ δ = { x ∈ Ω ; d ( x , ∂ Ω ) ≤ δ } .$

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

$f 0 = max { 0 , - f ( 0 ) } , h 0 = max { 0 , - h ( 0 ) } .$

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

$A = max [ γ λ p η a ( γ 1 p - 1 α p ) + f ( γ 1 q - 1 α q ) , γ λ q η b ( γ 1 q - 1 α q ) + h ( γ 1 p - 1 α p ) ] B = min [ m γ β a ( γ 1 p - 1 ) + f 0 , m γ β b ( γ 1 q - 1 ) + h 0 ]$

where $α p = p - 1 p μ p p p - 1$ and $α q = q - 1 q μ q q q - 1$.

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 3.1 Suppose 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

$ψ 1 = γ 1 p - 1 p - 1 p φ p p p - 1 ψ 2 = γ 1 q - 1 q - 1 q φ q q q - 1 .$

Let $W∈ H 0 1 ( Ω )$ with w ≥ 0. Then

$∫ Ω | ∇ ψ 1 | p - 2 ∇ ψ 1 ∇ w d x = γ ∫ Ω ( λ p φ p p - | ∇ φ p | p ) w d x$
(4)

Now, on $Ω ¯ δ$ by (2),(3) we have

$γ ( λ p φ p p - | ∇ φ p | p ) ≤ - m γ$

Since λB then

$λ ≤ m γ β a ( γ 1 p - 1 ) + f 0 .$

thus

then by (4)

A similar argument shows that

Next, on $Ω- Ω ¯ δ$. Since λA, then

$λ ≥ γ λ p η a γ 1 p - 1 α p + f γ 1 q - 1 α q$

so we have

$γ ( λ p φ p p - | ∇ φ p | p ) ≤ γ λ p ≤ λ η a γ 1 p - 1 α p + f γ 1 q - 1 α q ≤ λ [ g ( x ) a ( ψ 1 ) + f ( ψ 2 ) ] , Ω - Ω δ ¯$

Then by (4) on we have

$- Δ p ψ 1 ≤ λ [ g ( x ) a ( ψ 1 ) + f ( ψ 2 ) ] on Ω - Ω δ ¯$

A similar argument shows that

$- Δ q ψ 2 ≤ λ [ g ( x ) b ( ψ 2 ) + h ( ψ 1 ) ]$

We suppose that κ p and κ q be solutions of

$- Δ p κ p = 1 in Ω κ p = 0 on ∂ Ω - Δ q κ q = 1 in Ω κ q = 0 on ∂ Ω$

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

Let

$( z 1 , z 2 ) = c μ ′ p λ 1 p - 1 κ p , 2 h c λ 1 q - 1 1 q - 1 λ 1 q - 1 κ q .$

Let $W∈ H 0 1 ( Ω )$ with w ≥ 0.

For sufficient C large

$μ ′ p p - 1 [ | | g | | ∞ a ( C λ 1 p - 1 ) + f ( ( 2 h ( C λ 1 p - 1 ) ) 1 q - 1 λ 1 q - 1 μ ′ q ) ] C p - 1 ≤ 1$

then

$∫ | ∇ z 1 | p - 2 ∇ z 1 ∇ w d x = λ C μ ′ p p - 1 ∫ w d x ≥ λ ∫ | | g | | ∞ a ( C λ 1 p - 1 ) + f ( 2 h ( C λ 1 p - 1 ) ) 1 q - 1 λ 1 q - 1 μ ′ q d x ≥ λ ∫ g ( x ) a ( C λ 1 p - 1 κ p μ ′ p ) + f ( 2 h ( C λ 1 p - 1 ) ) 1 q - 1 λ 1 q - 1 κ q d x = ∫ [ g ( x ) a ( z 1 ) + f ( z 2 ) ] w d x$

Similarly, choosing C large so that

then

$∫ | ∇ z 2 | q - 2 ∇ z 2 ∇ w d x = 2 λ h C λ 1 p - 1 ∫ w d x ≥ λ ∫ | | g | | ∞ b ( z 2 ) + h ( z 1 ) w d x .$

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

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## Author information

Correspondence to Somayeh Haghaieghi.

### 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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• #### DOI

https://doi.org/10.1186/1687-2770-2011-52

### Keywords

• Boundary Condition
• Differential Equation
• Partial Differential Equation
• Ordinary Differential Equation
• Positive Constant