Isolated singularities of positive solutions of elliptic equations with weighted gradient term

Abstract

Let $\Omega \subset {ℝ}^N$ ($N>2$) be a $C^2$ bounded domain containing the origin $0$. We study the behavior near $0$ of positive solutions of equation (E) $-\Delta u+|x{|}^{\alpha }{u}^p+|x{|}^{\beta }|\nabla u{|}^q=0$ in $\Omega \setminus \left\{0\right\}$, where $\alpha >-2$, $\beta >-1$, $p>1$, and $q>1$. When $1<p<\left(N+\alpha \right)∕\left(N-2\right)$ and $1<q<\left(N+\beta \right)∕\left(N-1\right)$, we provide a full classification of positive solutions of (E) vanishing on $\partial \Omega$. On the contrary, when $p\ge \left(N+\alpha \right)∕\left(N-2\right)$ or $\left(N+\beta \right)∕\left(N-1\right)\le q\le 2+\beta$, we show that any isolated singularity at $0$ is removable.