Positive solutions of an elliptic equation with negative exponent: stability and critical power

Du, Yihong; Guo, Zongming

doi:10.1016/j.jde.2008.08.008

Author(s)

Du, Yihong

Guo, Zongming

Publication Date

2009

Abstract

We study positive solutions of the equation Δu = |x|α u−p in Ω ⊂ RN (N ≥ 2), where p > 0, α > −2, and Ω is a bounded or unbounded domain. We show that there is a critical power p = pc(α) such that this equation with Ω = ℝN has no stable positive solution for p > pc(α) but it admits a family of stable positive solutions when 0 < p ≤ pc(α). If p > pc(α⁻) (α⁻ = min{α, 0}), we further show that this equation with Ω = Br {0} has no positive solution with finite Morse index that has an isolated rupture at 0, and analogously it has no positive solution with finite Morse index when Ω = ℝN BR . Among other results, we also classify the positive solutions over Br {0} which are not bounded near 0.

Citation

Journal of Differential Equations, 246(6), p. 2387-2414

ISSN

1090-2732

0022-0396

Link

https://hdl.handle.net/1959.11/6228

Language

en

Publisher

Academic Press

Title

Positive solutions of an elliptic equation with negative exponent: stability and critical power

Type of document

Journal Article

Entity Type

Publication

Author(s)	Du, Yihong Guo, Zongming
Publication Date	2009
Abstract	We study positive solutions of the equation Δu = \|x\|α u−p in Ω ⊂ RN (N ≥ 2), where p > 0, α > −2, and Ω is a bounded or unbounded domain. We show that there is a critical power p = pc(α) such that this equation with Ω = ℝN has no stable positive solution for p > pc(α) but it admits a family of stable positive solutions when 0 < p ≤ pc(α). If p > pc(α⁻) (α⁻ = min{α, 0}), we further show that this equation with Ω = Br {0} has no positive solution with finite Morse index that has an isolated rupture at 0, and analogously it has no positive solution with finite Morse index when Ω = ℝN BR . Among other results, we also classify the positive solutions over Br {0} which are not bounded near 0.
Citation	Journal of Differential Equations, 246(6), p. 2387-2414
ISSN	1090-2732 0022-0396
Link	https://hdl.handle.net/1959.11/6228
Language	en
Publisher	Academic Press
Title	Positive solutions of an elliptic equation with negative exponent: stability and critical power
Type of document	Journal Article
Entity Type	Publication

Positive solutions of an elliptic equation with negative exponent: stability and critical power

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