Aleksandrov-Bakelman-Pucci maximum principle for $L^n$-viscosity solutions of uniformly elliptic equations with unbounded terms: The critical case

Abstract

We prove the Aleksandrov-Bakelman-Pucci maximum principle for $L^n$-viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations in $\Omega \subset {\mathbb{R}}^n$ with the drift coefficient $\gamma$ such that $\gamma (\ln (1+\gamma ){)}^{\frac{n-1}{n}}\in L^n(\Omega )$. The approach is based on convex duality and Trudinger-Moser type inequalities.