Abstract
In this work, we consider the system:
where Ω is a bounded region in R^{N }with smooth boundary ∂Ω, Δ_{p }is the pLaplacian operator defined by Δ_{p}u = div (∇u^{p2}∇u), p, q > 1 and g (x) is a C^{1 }signchanging the weight function, that maybe negative near the boundary. f, h, a, b are C^{1 }nondecreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0. Using the method of subsuper 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 R^{N }with smooth boundary ∂Ω, Δ_{p }is the pLaplacian operator defined by Δ_{p}u = div(∇u^{p2 }∇u), p, q > 1 and g(x) is a C^{1 }signchanging the weight function, that maybe negative near the boundary. f, h, a, b are C^{1 }nondecreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0.
This paper is motivated by results in [15]. We shall show the system (1) with signchanging weight functions has at least one solution.
2 Preliminaries
In this article, we use the following hypotheses:
(Al)
(A2) lim f (s) = lim h (s) = ∞ as s → ∞.
(A3)
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 ΩΩ_{δ}. To be precise, we assume that there exist positive constants β and η such that g(x) ≥β on
For γ such that γ^{r1 }t > s_{0}; t = min {α_{p}, α_{q}}, r = min{p, q} we define
where
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})].
3 Main result
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 (ψ_{1}, ψ_{2}) is a sub solution of (1.1) where
Let
Now, on
Since λ ≤ B then
thus
then by (4)
A similar argument shows that
Next, on
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
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 (ψ_{1}, ψ_{2}) ≤ (u, v) ≤ (z_{1}, z_{2}).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
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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