Global $L^\infty$-estimate for general quasilinear elliptic equations in arbitrary domains of $\mathbb{R}^N$

Abstract

In this paper our main goal is to present a new global $L^\infty$-estimate for a general class of quasilinear elliptic equations of the form $$ -div \mathcal{A}(x,u,\nabla u)=\mathcal{B}(x,u,\nabla u) $$ under minimal structure conditions on the functions $\mathcal{A}$ and $\mathcal{B}$, and in arbitrary domains of $\mathbb{R}^N$. The main focus and the novelty of the paper is to prove $L^\infty$-estimate of the form $$ |u|_{\infty, \Omega}\le C \Phi(|u|_{\beta,\Omega}) $$ where $\Phi: \mathbb{R}^+\to \mathbb{R}^+$ is a data independent function with $\lim_{s\to 0^+}\Phi(s)=0$.