A nonlinear elliptic system with a transport term and singular data

Abstract

We consider the nonlinear elliptic system { u ∈ W 0 N N − 1 ( Ω ) : − div ( M ( x ) ∇u ) + u = − div ( u M ( x ) ∇ψ ) + f ( x ) , ψ ∈ W 0 1 , 2 ( Ω ) : − div ( M ( x ) ∇ψ ) + ψ = R ( u ) + E ( x ) ∇ψ , where Ω is a bounded, open subset of R N , N ≥ 3 ; M ( x ) is a coercive, symmetric matrix with L ∞ ( Ω ) coefficients; f ( x ) and E ( x ) belong to some Lebesgue space, and R ( s ) is a continuous function such that 0 ≤ R ( s ) ≤ | s | θ , for θ < 2 N . Using a duality technique, we prove existence of at least a weak solution ( u , ψ ) . Moreover, if N=3 or N=4, we prove under stronger assumptions on f ( x ) and E ( x ) that the solution u belongs to W 0 1 , 2 ( Ω ) .