Renormalized solutions in anisotropic weighted sobolev spaces: Existence and uniqueness results

Abstract

This paper focuses on the study of the following elliptic problem:$ \left\{\begin{array}{ll} - \displaystyle\sum\limits_{i = 1}^{N}D^{i} \left( a_{i}(x, \nabla u) + \phi_{i}(x,u) \right) = f & \mbox{ in } \ \ \Omega, \\ \\ u = 0 & \mbox{ on } \ \ \partial\Omega. \end{array}\right. $We establish the existence and uniqueness of renormalized solutions in anisotropic weighted Sobolev spaces. Additionally, we prove some regularity results.