$L^\infty$-bounds for general singular elliptic equations with convection term

TL;DR

This paper establishes $L^ abla$ bounds for positive solutions to a broad class of singular elliptic equations with convection terms, extending existing methods to handle combined singular and convection growth conditions.

Contribution

It introduces new $L^ abla$-boundedness results for solutions of singular elliptic equations with convection, using bootstrap techniques adapted to complex growth conditions.

Findings

Proved boundedness of weak solutions under combined singular and convection growth conditions.

Extended bootstrap methods to handle general singular elliptic problems.

Provided a framework for $L^ abla$-bounds in complex elliptic PDEs.

Abstract

In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where the growth condition for the function $B\colon \Omega \times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.