L^infinity solutions for anisotropic singular elliptic equation with convection term

On Bounded Solutions for Anisotropic Singular Elliptic Equation

Abstract

In this paper we will prove the boundedness of weak solutions of nonlinear anisotropic elliptic equations of the form \begin{equation*} \begin{cases} -\sum_{i=1}^{N}D_{i}a_{i}(x,u,\nabla u))=\sum_{i=1}^{N}b_{i}(x,u,\nabla u) & \text{in } \Omega,\\ u> 0&\text{in } \Omega,\\ u= 0&\text{on } \partial\Omega,\\ \end{cases} \end{equation*} where the growth condition for the functions $b_{i} : \Omega\times\mathbb{R}\times \mathbb{R}^{N}\rightarrow\mathbb{R}$ for all $i=1,\ldots,N$ contains the singular terms and a convection term. The core concept in the proof relies on an adapted iteration technique inspired by Moser's approach. This work generalizes some results given in \cite{9}.