Calculus of Variations and Geometric Measure Theory

M. Cirant - A. Goffi

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

created by goffi on 29 Jul 2021
modified on 01 Feb 2022


Accepted Paper

Inserted: 29 jul 2021
Last Updated: 1 feb 2022

Journal: Contemporary Mathematics, AMS, Proceedings of the conference "Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs"
Year: 2021

ArXiv: 2107.13262 PDF


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 $H_1(u,Du)=u^q+
^\gamma$, $H_2(u,Du)=u^q
^\gamma$ and $H_3(x,Du)=\pm u^q
^\gamma-b(x)\cdot Du$, where $q\geq0$, $\gamma>1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.

Keywords: Hamilton-Jacobi equations, Liouville theorems, Fully nonlinear equation, Lane-Emden equations, superlinear gradient terms