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

TL;DR

This paper investigates positive solutions of nonlinear elliptic equations with absorption depending on the solution and its gradient, establishing existence, uniqueness, and classification results based on conditions on parameters p and q.

Contribution

It provides sharp conditions for existence and removability of boundary singularities, and classifies solutions with isolated boundary singularities for specific nonlinear elliptic equations.

Findings

Existence of solutions for given boundary measures under certain parameter conditions.

Removability of isolated boundary singularities when conditions fail.

Classification of solutions with boundary singularities and solutions that blow up on subsets.

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.