# A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

## TL;DR

This paper proves that all positive bounded solutions to a certain elliptic PDE with superquadratic gradient growth are constant, using spherical averages and gradient bounds, extending previous results to the case p ≥ 2.

## Contribution

It establishes a Liouville-type theorem for elliptic equations with superquadratic gradient growth, filling a gap for the case p ≥ 2.

## Key findings

- Positive bounded solutions are constant.
- The proof uses spherical averages and Bernstein gradient bounds.
- Extends previous results to new parameter range p ≥ 2.

## Abstract

We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.