Local boundedness for $p$-Laplacian with degenerate coefficients

TL;DR

This paper establishes local boundedness for solutions of a class of nonlinear elliptic equations with unbounded coefficients, identifying optimal integrability conditions and providing counterexamples to regularity.

Contribution

It introduces new integrability conditions on coefficients ensuring local boundedness and demonstrates their optimality with counterexamples.

Findings

Local boundedness holds under specific integrability conditions.

Counterexamples show the conditions are sharp and cannot be weakened.

Results extend regularity theory to equations with degenerate coefficients.

Abstract

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leq\lambda$ and its inverse $\lambda^{-1}$ are allowed to be unbounded. Assuming certain integrability conditions on $\lambda$ and $\lambda^{-1}$ depending on $p$ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $p>1$.