On a Nonlinear System of Reaction-Diffusion Equations

Abstract

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,
− div(|∇u₁|^p₁−2∇u₁) = λu₁^α₁₁u₂^α₁₂... u_n^α_1n,   x ∈ Ω,
− div(|∇u₂|^p₂−2∇u₂) = λu₁^α₂₁u₂^α₂₂... u_n^α_2n,   x ∈ Ω, ... , 
− div(|∇u_n|^p_n−2∇u_n) = λu₁^α_n1u₂^α_n2... u_n^α_nn,   x ∈ Ω,
u_i = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n, 

where λ is a positive parameter, Ω is a bounded domain in R^N (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < p_i < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.