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

In this work, we consider the system:

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 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.

In this paper, we study the existence of positive weak solution for the following system:

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 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.

In this article, we use the following hypotheses:

(Al) as s → ∞, ∀M > 0

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

(A3) 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

and

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 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

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

where and .

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)].

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

Let with w ≥ 0. Then

Now, on by (2),(3) we have

Since λ ≤ B then

thus

then by (4)

A similar argument shows that

Next, on . Since λ ≥ A, then

so we have

Then by (4) on we have

A similar argument shows that

We suppose that κ_p and κ_q be solutions of

respectively, and μ'_p = ||κ_p||_κ, ||κ_q||_κ = μ'_q.

Let

Let with w ≥ 0.

For sufficient C large

then

Similarly, choosing C large so that

then

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

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.

1. Ali, J, Shivaji, R: Existence results for classes of Laplacian system with sign-changing weight. Appl Math Anal. 20, 558–562 (2007)

2. Rasouli, SH, Halimi, Z, Mashhadban, Z: A remark on the existence of positive weak solution for a class of (p, q)-Laplacian nonlinear system with sign-changing weight. Nonlinear Anal. 73, 385–389 (2010). Publisher Full Text

3. Ali, J, Shivaji, R: Positive solutions for a class of (p)-Laplacian systems with multiple parameters. J Math Anal Appl. 335, 1013–1019 (2007). Publisher Full Text

4. Hai, DD, Shivaji, R: An existence results on positive solutions for class of semilinear elliptic systems. Proc Roy Soc Edinb A. 134, 137–141 (2004). Publisher Full Text

5. Hai, DD, Shivaji, R: An Existence results on positive solutions for class of p-Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004). Publisher Full Text

6. Canada, A, Drabek, P, Azorero, PL, Peral, I: Existence and multiplicity results for some nonlinear elliptic equations. A survey Rend Mat Appl. 20, 167–198 (2000)