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Sub-super solutions for (p-q) Laplacian systems
Boundary Value Problems volume 2011, Article number: 52 (2011)
Abstract
In this work, we consider the system:
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-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.
1 Content
In this paper, we study the existence of positive weak solution for the following system:
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-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.
2 Preliminaries
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 γr-1t > 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)].
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
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).
References
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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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Haghaieghi, S., Afrouzi, G.A. Sub-super solutions for (p-q) Laplacian systems. Bound Value Probl 2011, 52 (2011). https://doi.org/10.1186/1687-2770-2011-52
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DOI: https://doi.org/10.1186/1687-2770-2011-52