Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

TL;DR

This paper proves that all positive solutions to a certain quasilinear elliptic equation in the entire space are constant, and describes their asymptotic behavior, using gradient estimates and Liouville-type theorems.

Contribution

It establishes a Liouville-type result for positive solutions of a supercritical quasilinear elliptic equation and analyzes their asymptotic properties.

Findings

All positive solutions in R^N are constant.

Solutions in exterior domains are bounded.

Solutions near the origin extend continuously and have finite limits at infinity.

Abstract

We consider the elliptic quasilinear equation --$\Delta$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman's type estimate for the equation satisfied by the gradient.