Regularity of solutions of quasi-linear elliptic equations with $L\log^m L$ coefficients

TL;DR

This paper investigates the regularity of solutions to certain quasi-linear elliptic equations with coefficients in the space $L ext{log}^m L$, establishing local boundedness, Harnack inequalities, and continuity results that are sharp with respect to the parameter $m$.

Contribution

It introduces new regularity results for solutions with coefficients in $L ext{log}^m L$, using Moser iteration and Moser-Trudinger inequality, extending previous understanding of elliptic equations.

Findings

Established local $L^{ abla}$ bounds on solutions.

Proved Harnack-type inequalities for solutions.

Showed regularity results are sharp with respect to $m$.

Abstract

Let $D$ be an bounded region in ${\bf R}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being $\Delta_nu=Vu^{n-1}$. The coefficients are assumed to be in the space $L\log^{m}L(D)$ for $m>n-1$. Using a Moser iteration argument coupled with the Moser-Trudinger inequality, a local $L^{\infty}$ bound on the solution $u$ is proven. A Harnack-type inequality is then proven. These results are shown to be sharp with respect to $m$. Then essential continuity of $u$ is proven, and away from the boundary a bound on the modulus of continuity.