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 ,

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

This paper is motivated by results in 1 2 3 4 5 . We shall show the system (1) with sign-changing weight functions has at least one solution.

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 = l i m 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

and

{ | ∇ φ q | q - λ q φ q ≥ m in Ω ¯ δ φ p ≥ μ p on Ω - Ω δ .

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 s ₀ ≥ 0 such that ηa(s) + f (s) > 0, ηb(s) + h(s) > 0 for s > s ₀ and

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

For γ such that γ ^r-1 t > s ₀; 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 (ψ ₁, ψ ₂) and (z ₁, z ₂) respectively of (1) such that (ψ ₁, ψ ₂) ≤ (z ₁, z ₂). then (1) has a solution (u, v) such that (u, v) ∈ [(ψ ₁, ψ ₂), (z ₁, z ₂)].

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 (ψ ₁, ψ ₂) 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

Now, on Ω ¯ δ by (2),(3) we have

γ ( λ p φ p p - | ∇ φ p | p ) ≤ - m γ

Since λ ≤ B 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 λ ≥ A, 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 (ψ ₁, ψ ₂) ≤ (u, v) ≤ (z ₁, z ₂).

The authors declare that they have no competing interests.

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.