On the Liouville property for fully nonlinear equations with superlinear first-order terms

TL;DR

This paper investigates the conditions under which viscosity solutions of fully nonlinear elliptic inequalities with superlinear gradient growth exhibit Liouville properties, providing new insights and counterexamples in the field.

Contribution

It analyzes the validity of Liouville properties for specific superlinear inequalities, including new counterexamples and open problems in the theory.

Findings

Liouville property holds under certain conditions for superlinear inequalities

Counterexamples demonstrate failure of Liouville property in some cases

Open problems identified for further research in nonlinear elliptic equations

Abstract

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is $F(x,D^2u)\geq H_i(x,u,Du)$ in $\mathbb{R}^N$, where $H_i$ has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases $H_1(u,Du)=u^q+|Du|^\gamma$, $H_2(u,Du)=u^q|Du|^\gamma$ and $H_3(x,u,Du)=\pm u^q|Du|^\gamma-b(x)\cdot Du$, where $q\geq0$, $\gamma>1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.