Calderon–Zygmund–Stampacchia theory for infinite energy solutions of nonlinear elliptic equations with singular drift

Abstract

In this paper we study the existence and regularity of solutions to some nonlinear boundary value problems with non coercive drift. The model problem is

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(A(x)\nabla u|\nabla u|^{p-2} )=E(x)\nabla u|\nabla u|^{p-2}+f(x), &{} \text {in } \Omega ; \\ u =0, &{} \text {on } \partial \Omega ; \end{array}\right. \end{aligned}$$

(1)

where $p>1$, $\Omega$ is an open bounded subset of ${\mathbb {R}}^N$, A(x) is an elliptic matrix with measurable and bounded entries,$E\in (L^{N}(\Omega ))^N$ and $f\in L^{m}(\Omega )$ with $1<m<\frac{N}{p}$. No further regularity on the coefficients of A(x) is used and no smallness assumption of $\Vert |E|\Vert _{L^{N}(\Omega )}$ is required. Our strategy is based on the proof of a priori estimates by contradiction.