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

We study positive solutions of equation (E1) $-Δu + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-Δu + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $Ω\subset \mathbb{R}^N$. We obtain a sharp condition on $p$ and $q$ under which, for every positive, finite Borel measure $μ$ on $\partial Ω$, there exists a solution such that $u=μ$ on $\partial Ω$. Furthermore, if the condition mentioned above fails then any isolated point singularity on $\partial Ω$ is removable, namely there is no positive solution that vanishes on $\partial Ω$ everywhere except at one point. With respect to (E2) we also prove uniqueness and discuss solutions that blow-up on a compact subset of $\partial Ω$. 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.

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