An elliptic system involving a singular diffusion matrix

Abstract

Let Ω ⊂ R N ( N > 1 ) be a bounded domain. In this work we are interested in finding a renormalized solution to the following elliptic system (1) { − div [ A 1 ( u 2 ) ∇ u 1 ] = f , in Ω − div [ A 2 ( u 2 ) ∇ u 2 ] + g ( u 2 ) = A 3 ( u 2 ) ∇ u 1 ∇ u 1 , in Ω , where the diffusion matrix A 2 blows up for a finite value of the unknown, say u 2 = s 0 < 0 . We also consider homogeneous Dirichlet boundary conditions for both u 1 and u 2 . In these equations, u 1 is an N -dimensional magnitude, whereas u 2 is scalar; A 2 : Ω × ( s 0 , + ∞ ) ↦ R N is a semilinear coercive operator. The symmetric part of the matrix A 3 is related to the one of A 1 . Nevertheless, the behaviour of these coefficients is assumed to be fairly general. Finally, f ∈ H − 1 ( Ω ) N , and g : Ω × ( s 0 , + ∞ ) ↦ 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.