Positive solutions of quasilinear elliptic equations with subquadratic growth in the gradient

TL;DR

This paper investigates positive solutions of a class of quasilinear elliptic equations with subquadratic gradient growth, establishing existence, removability of singularities, and classification of boundary singular solutions based on domain and parameter conditions.

Contribution

It provides new existence and non-existence results for positive solutions with boundary measures, and classifies solutions with isolated boundary singularities under specific conditions.

Findings

Existence of solutions for any positive finite boundary measure when N(p+q-1)<p+1.

Removability of isolated boundary singularities when N(p+q-1)≥p+1.

Complete classification of positive solutions with isolated boundary singularities.

Abstract

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for every positive, finite Borel measure $\mu$ on $\partial \Omega$, there exists a solution of (E) such that $u=\mu$ on $\partial \Omega$. Furthermore, if $N(p+q-1)\geq p+1$ then an isolated point singularity on $\partial \Omega$ is removable. In particular there is no solution with boundary data $\delta_y$ (=Dirac measure at a point $y\in \partial \Omega$). Finally we obtain a classification of positive solutions with an isolated boundary singularity.