#### Vol. 9, No. 7, 2016

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

### Phuoc-Tai Nguyen

Vol. 9 (2016), No. 7, 1671–1692
DOI: 10.2140/apde.2016.9.1671
##### 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 and $1, 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.

##### Keywords
gradient terms, weak singularities, strong singularities, removability
##### Mathematical Subject Classification 2010
Primary: 35A20, 35J60