Sub-super solutions for (p-q) Laplacian systems
© Haghaiegh and Afrouzi; licensee Springer. 2011
Received: 13 August 2011
Accepted: 2 December 2011
Published: 2 December 2011
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 p u = 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.
where Ω is a bounded region in R N with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = 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.
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.
where and .
We use the following lemma to prove our main results.
Lemma 1.1. 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.
respectively, and μ' p = ||κ p ||κ, ||κ q ||κ = μ' q .
Let with w ≥ 0.
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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