existence and nonexistence results for an ecological model with indefinite weight

This study concerns the existence of a positive solution for the followingnon 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.