On an elliptic chemotaxis system with flux limitation and subcritical signal production

Abstract

In this article we study the existence of solutions of a system of partial differential equations of elliptic type, describing the distribution of a biological species “u” and the density of a chemical stimulus “ψ” in a bounded domain Ω of RN. The equation for u includes a chemotaxis term with nonlinear flux limitation which depends on the exponent p>1.The equation for u is given by −div(M(x)∇u)+u=−χdiv(u|∇ψ|p−2∇ψ)+f(x),where ψ presents a subcritical production term uθ and satisfies the equation −div(M(x)∇ψ)+ψ=uθ.The matrix of coefficients, M, is a known, symmetric and positive defined with coefficients mij∈C1(Ω¯),χ is a given real constant, f is a non-negative function belonging to Lm(Ω), m>max{1,N2}. The production term exponent, θ, is assumed to be positive and fulfills one of the following constrains 1<p<NθNθ−1,1<Nθor max{N,p}<θ+1θ for θ>0.The problem is completed with Dirichlet boundary conditions for u and ψ. The main result of the article includes the existence of positive solutions in H01(Ω)∩L∞(Ω).