On the Existence, Nonexistence and Uniqueness of Positive Weak Solutions for Nonlinear Multiparameter Elliptic Systems Involving the (P,Q)-Laplacian

The paper deal with the existence, nonexistence and uniqueness of positive weak solutions for the nonlinear multiparameter elliptic system

{ −Δ_pu = λ₁ f(v) + μ1 h(u); x∈Ω, −Δ_qv = λ2 g(u) + μ₂ k(v); x ∈ Ω, u=0=v; x ∈ ∂Ω,

where Ω is a bounded domain in R^N(N > 1) with smooth boundary ∂Ω, Δp denotes the p-Laplacian operator defined by Δ_pz = div (|∇_z|^p-2􀀀∇_z), p>1,f, g, h, k: [0,∞) → [0,∞), and λ₁, λ₂, μ₁,μ₂ are positive parameters. A positive weak solution is obtained by applying the Schauder Fixed Point Theorem, together with comparison principles for the p-Laplacian. A uniqueness and nonexistence result is also obtained.