We consider the elliptic equation -Delta u = u^q|nabla u|^p in R^n for any p>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 a recent paper, where the authors consider the case 0<2. Some extensions to elliptic systems are also given.

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

Filippucci, Roberta;Pucci, Patrizia;
2020

Abstract

We consider the elliptic equation -Delta u = u^q|nabla u|^p in R^n for any p>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 a recent paper, where the authors consider the case 0<2. Some extensions to elliptic systems are also given.
2020
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/1452031
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