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

Abstract

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