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Research | Open | Published:

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

In this article, we use the following hypotheses:

(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

γ ( λ p φ p p - | φ p | p ) - m γ λ - β a ( γ 1 p - 1 ) - f 0 λ g ( x ) a ( γ 1 p - 1 ) - f 0 λ g ( x ) a p - 1 p γ 1 p - 1 φ p 1 p - 1 + f q - 1 q γ 1 q - 1 φ q 1 q - 1

then by (4)

Ω ¯ δ | ψ 1 | p - 2 ψ 1 w d x Ω ¯ δ λ g ( x ) a p - 1 p γ 1 p - 1 φ p p p - 1 + f q - 1 q γ 1 q - 1 φ q q q - 1 w d x

A similar argument shows that

Ω ¯ δ | ψ 2 | q - 2 ψ 2 w d x Ω ¯ δ λ g ( x ) b q - 1 q γ 1 q - 1 φ q 1 q - 1 + h p - 1 p γ 1 p - 1 φ p 1 p - 1 w d x

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

| | g | | b 2 h C λ 1 p - 1 1 q - 1 λ 1 q - 1 μ q h C λ 1 p - 1 1

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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Correspondence to Somayeh Haghaieghi.

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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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Keywords

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