A nonlinear elliptic system with a transport term and singular data

In this article, we study a nonlinear system of elliptic partial differential equations describing the interaction of two species, “ u ” and “ ψ ” in a bounded domain Ω of R N for N ≥ 3 . The equation for “ ψ ” presents a production term defined by a bounded function B ( u ) and a drift term, which depends on a known function E . The system is presented in the following way model problem u ∈ W 0 1 , N N - 1 ( Ω ) : - div ( A ( x ) ∇ u ) + u = - div ( u ∇ ψ ) + f , ψ ∈ W 0 1 , 2 ( Ω ) : - div ( A ( x ) ∇ ψ ) + ψ = B ( u ) + E ∇ ψ . where A : Ω → R N 2 is a symmetric matrix with bounded coefficients a ij for i , j = 1 ⋯ N , and E : Ω → R N belongs to ( L N ( Ω ) ) N , f is assumed to be a non-negative function of L 1 ( Ω ) satisfying ∫ Ω f log ( 1 + f ) < ∞ and B is a continuous and bounded function. We obtain the existence of solutions of the model problem, moreover, for N = 3 , if f ∈ L ∞ ( Ω ) and E ∈ L ∞ ( Ω ) 3 we have that u ∈ W 0 1 , 2 ( Ω ) ∩ L ∞ ( Ω ) .