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

Abstract

Let $D$ be an bounded region in ${\mathbf{R}}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being ${\Delta }_{n}u=V{u}^{n-1}$. 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.