An elliptic problem in dimension N with a varying drift term bounded in $L^N$

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^N(\mathrm{\Omega })$, with $N$ the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space $H_0^1(\mathrm{\Omega })$. However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in \cite{boc1}, and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.