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arXiv · 1907.06816

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

Roberta Filippucci·Patrizia Pucci·Philippe Souplet
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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.

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Roberta Filippucci, Patrizia Pucci, Philippe Souplet. 2019-07-16. A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient. https://arxiv.org/abs/1907.06816

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