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).
- Ali J, Shivaji R: Existence results for classes of Laplacian system with sign-changing weight. Appl Math Anal 2007, 20: 558-562.MathSciNetGoogle Scholar
- 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 2010, 73: 385-389. 10.1016/j.na.2010.03.027View ArticleMathSciNetGoogle Scholar
- Ali J, Shivaji R: Positive solutions for a class of (p)-Laplacian systems with multiple parameters. J Math Anal Appl 2007, 335: 1013-1019. 10.1016/j.jmaa.2007.01.067View ArticleMathSciNetGoogle Scholar
- Hai DD, Shivaji R: An existence results on positive solutions for class of semilinear elliptic systems. Proc Roy Soc Edinb A 2004, 134: 137-141. 10.1017/S0308210500003115View ArticleMathSciNetGoogle Scholar
- Hai DD, Shivaji R: An Existence results on positive solutions for class of p-Laplacian systems. Nonlinear Anal 2004, 56: 1007-1010. 10.1016/j.na.2003.10.024View ArticleMathSciNetGoogle Scholar
- Canada A, Drabek P, Azorero PL, Peral I: Existence and multiplicity results for some nonlinear elliptic equations. A survey Rend Mat Appl 2000, 20: 167-198.Google Scholar