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

Nguyen, Phuoc-Tai

doi:10.2140/apde.2016.9.1671

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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 $Ω \subset ℝ^{N}$ ( $N > 2$ ) be a $C^{2}$ bounded domain containing the origin $0$ . We study the behavior near $0$ of positive solutions of equation (E) $- Δ u + | x |^{α} u^{p} + | x |^{β} | \nabla u |^{q} = 0$ in $Ω ∖ {0}$ , where $α > - 2$ , $β > - 1$ , $p > 1$ , and $q > 1$ . When $1 < p < (N + α) ∕ (N - 2)$ and $1 < q < (N + β) ∕ (N - 1)$ , we provide a full classification of positive solutions of (E) vanishing on $\partial Ω$ . On the contrary, when $p \geq (N + α) ∕ (N - 2)$ or $(N + β) ∕ (N - 1) \leq q \leq 2 + β$ , 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

Milestones

Received: 20 October 2015

Revised: 21 April 2016

Accepted: 6 June 2016

Published: 7 November 2016

Authors

Phuoc-Tai Nguyen
Facultad de Matemáticas Pontificia Universidad Católica de Chile Avenida Vicuña Mackenna 4860 6904441 Santiago Chile

Vol. 9, No. 7, 2016