On the existence of positive solutions for an ecological model with indefinite weight

This study concerns the existence of positive solutions for the following nonlinear boundary value problem: {−Δu=am(x)u−bu2−cupup+1−Kin  Ω,u=0on  ∂Ω, where Δu=div(∇u)Δu=div(∇u) is the Laplacian of uu, while a,b,c,p,Ka,b,c,p,K are positive constants with p≥2p≥2 and ΩΩ is a bounded smooth domain of RNRN with ∂Ω∂Ω in C2C2. The weight function mm satisfies m∈C(Ω)m∈C(Ω) and m(x)≥m0>0m(x)≥m0>0 for x∈Ωx∈Ω, also ‖m‖∞=l<∞‖m‖∞=l<∞. We prove the existence of positive solutions under certain conditions.