*Submitted Paper*

**Inserted:** 29 jul 2021

**Last Updated:** 29 jul 2021

**Year:** 2021

**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,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.

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