EXISTENCE AND NONEXISTENCE RESULTS FOR AN ECOLOGICAL MODEL WITH INDEFINITE WEIGHT

This study concerns the existence of a positive solution for the following non linear boundary value problem −Δpu = am(x)u^p−1 − bu² − c u^γ/ u^γ+1 − K in Ω, u = 0 on ∂Ω. Here, Δpu := div(|∇u|^p−2∇u) is the p-Laplacian operator, p 1, a, b, c, γ,K are positive constants with γ ≥ 2, and Ω is a smooth bounded region with ∂Ω belonging to C². The weight function m(x) satisfies m(x) ∈ C(Ω) and m(x) ≥ m0 0 for x ∈ Ω, also m ∞ = l ∞. We prove the existence of a positive solution under certain conditions.