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 $Ω\subset\mathbb R^n$ with the drift coefficient $γ$ such that $γ(\ln(1+γ))^{\frac{n-1}{n}}\in L^n(Ω)$. The approach is based on convex duality and Trudinger-Moser type inequalities.