Uniqueness of radially symmetric positive solutions for - Δ u + u = up in an annulus

Patricio Felmer; Salomé Martínez; Kazunaga Tanaka

doi:10.1016/j.jde.2008.06.006

Uniqueness of radially symmetric positive solutions for - Δ u + u = u^p in an annulus

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

22 Citations (Scopus)

Abstract

In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = u^p in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ R^N | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate.

Original language	English
Pages (from-to)	1198-1209
Number of pages	12
Journal	Journal of Differential Equations
Volume	245
Issue number	5
DOIs	https://doi.org/10.1016/j.jde.2008.06.006
Publication status	Published - 2008 Sept 1

Keywords

Non-degeneracy
Non-linear elliptic equation
Radially symmetric solutions

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jde.2008.06.006

Cite this

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title = "Uniqueness of radially symmetric positive solutions for - Δ u + u = up in an annulus",

abstract = "In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = up in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ RN | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate.",

keywords = "Non-degeneracy, Non-linear elliptic equation, Radially symmetric solutions",

author = "Patricio Felmer and Salom{\'e} Mart{\'i}nez and Kazunaga Tanaka",

note = "Funding Information: The authors were partially supported by Fondecyt #1030929 and FONDAP de Matem{\'a}ticas Aplicadas (P.F.), Fondecyt #1050754, FONDAP de Matem{\'a}ticas Aplicadas and Nucleus Millenium P04-069-F Information and Randomness (S.M.), Grant-in-Aid for Scientific Research (B) (No. 20340037) JSPS and Fondecyt #7060276 (K.T.)",

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doi = "10.1016/j.jde.2008.06.006",

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T1 - Uniqueness of radially symmetric positive solutions for - Δ u + u = up in an annulus

AU - Felmer, Patricio

AU - Martínez, Salomé

AU - Tanaka, Kazunaga

N1 - Funding Information: The authors were partially supported by Fondecyt #1030929 and FONDAP de Matemáticas Aplicadas (P.F.), Fondecyt #1050754, FONDAP de Matemáticas Aplicadas and Nucleus Millenium P04-069-F Information and Randomness (S.M.), Grant-in-Aid for Scientific Research (B) (No. 20340037) JSPS and Fondecyt #7060276 (K.T.)

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N2 - In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = up in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ RN | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate.

AB - In this article we prove that the semi-linear elliptic partial differential equation- Δ u + u = up in Ω,u > 0 in Ω, u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus {x ∈ RN | a < | x | < b}, with N ≥ 2, 0 < a < b ≤ ∞. We also show the positive solution is non-degenerate.

KW - Non-degeneracy

KW - Non-linear elliptic equation

KW - Radially symmetric solutions

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