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

Abstract

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is $F(x,D^{2}u)\ge H_i(x,u,Du)$ 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 $q\ge 0$, $\gamma >1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.

arXiv comment27 pages