Existence of positive weak solution for a weighted system of autocatalytic reaction steady state type

Abstract

We establish the existence results of positive weak solution for the weighted p-Laplacian autocatalytic reaction problem −∆ P,p u = λm(x)[νa(x)u α − υu β ] in Ω, u = 0 on ∂Ω, where ∆ P,p with p > 1 and P = P (x) is a weight function, denotes the weighted p-Laplacian defined by ∆ P,p u ≡ div[P (x)|∇u| p−2 ∇u], m(x), a(x) are weight functions, λ, ν, υ are positive parameters, α + 1 ≤ p < β + 1, and Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω. We establish that there exists positive constant λ * (Ω) such that the above system has a positive weak solution for λ ≥ λ * . We use the method of sub-supersolutions to establish our results.