An elliptic system involving a singular diffusion matrix

摘要

Let Ω⊂RN (N>1) be a bounded domain. In this work we are interested in finding a renormalized solution to the following elliptic system (1){−div[A1(u2)∇u1]=f,in Ω−div[A2(u2)∇u2]+g(u2)=A3(u2)∇u1∇u1,in Ω, where the diffusion matrix A2 blows up for a finite value of the unknown, say u2=s0<0. We also consider homogeneous Dirichlet boundary conditions for both u1 and u2. In these equations, u1 is an N-dimensional magnitude, whereas u2 is scalar; A2:Ω×(s0,+∞)↦RN is a semilinear coercive operator. The symmetric part of the matrix A3 is related to the one of A1. Nevertheless, the behaviour of these coefficients is assumed to be fairly general. Finally, f∈H−1(Ω)N, and g:Ω×(s0,+∞)↦R is a Carathéodory function satisfying the sign condition.Due to these assumptions, the framework of renormalized solutions for problem (1) is used and an existence result is then established.