Elliptic equations with nonlinear absorption depending on the solution and its gradient

Abstract

We study positive solutions of equation (E1) $-\Delta u + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-\Delta u + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $\Omega \subset \mathbb{R}^N$. We obtain a sharp condition on $p$ and $q$ under which, for every positive, finite Borel measure $\mu$ on $\partial \Omega$, there exists a solution such that $u=\mu$ on $\partial \Omega$. Furthermore, if the condition mentioned above fails then any isolated point singularity on $\partial \Omega$ is removable, namely there is no positive solution that vanishes on $\partial \Omega$ everywhere except at one point. With respect to (E2) we also prove uniqueness and discuss solutions that blow-up on a compact subset of $\partial \Omega$. In both cases we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when $p=0$ but not the general case.