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 Ω N (N > 2) be a C2 bounded domain containing the origin 0. We study the behavior near 0 of positive solutions of equation (E) Δu + |x|αup + |x|β|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 Ω. On the contrary, when p (N + α)(N 2) or (N + β)(N 1) q 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