On the study of some non-coercive elliptic equations in anisotropic weighted Sobolev spaces

The primary focus of this paper is to establish some properties and lemmas concerning weighted anisotropic Sobolev spaces. As an application, we consider the following strongly nonlinear and non-coercive elliptic equation $$\label{p1} \left\{\begin{array}{lll} \displaystyle - \sum_{i=1}^{N}D^{i} a_{i}(x,u,\nabla u) + g(x,u,\nabla u) = f(x,u) & \mbox{ in } \Omega, \\ \displaystyle u = 0 & \mbox{ on } \partial \Omega, \end{array}\right.$$ where the right-hand side $f(x,s)$ verifying only a growth condition in $L^{1}(\Omega).$ We study the existence and regularity of renormalized solutions for our elliptic equation in the anisotropic weighted Sobolev spaces.\\ \textbf{ Mathematics Subject Classification (2020).} 46E35, 35J20, 35J60.\\ \textbf{Key words and phrases :} Anisotropic weighted Sobolev spaces, degenerate coercivity, non-coercive elliptic problems, renormalized solution.