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

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.