Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space

Yen Lin Wu, Zhi-You Chen, Jann Long Chern, Yoshitsugu Kabeya

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this article, we consider the following semilinear elliptic equation on the hyperbolic space ΔHnu - λu + |u|p-1u = 0 on Hn n\{Q} where ΔH n is the Laplace-Beltrami operator on the hyperbolic space Hn = {(x 1, · · · xn, xn+1)|x 1 2 + · · · + xn 2 - x n+1 2 = -1}, n > 10, p > 1, λ > 0, and Q = (0, · · · 0, 1). We provide the existence and uniqueness of a singular positive "radial" solution of the above equation for big p (greater than the Joseph-Lundgren exponent, which appears if n > 10) as well as its asymptotic behavior.

Original languageEnglish
Pages (from-to)949-960
Number of pages12
JournalCommunications on Pure and Applied Analysis
Volume13
Issue number2
DOIs
Publication statusPublished - 2014 Mar 1

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Singular Solutions
Hyperbolic Space
Elliptic Equations
Existence and Uniqueness
Positive Radial Solutions
Laplace-Beltrami Operator
Semilinear Elliptic Equations
Asymptotic Behavior
Exponent

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space",
abstract = "In this article, we consider the following semilinear elliptic equation on the hyperbolic space ΔHnu - λu + |u|p-1u = 0 on Hn n\{Q} where ΔH n is the Laplace-Beltrami operator on the hyperbolic space Hn = {(x 1, · · · xn, xn+1)|x 1 2 + · · · + xn 2 - x n+1 2 = -1}, n > 10, p > 1, λ > 0, and Q = (0, · · · 0, 1). We provide the existence and uniqueness of a singular positive {"}radial{"} solution of the above equation for big p (greater than the Joseph-Lundgren exponent, which appears if n > 10) as well as its asymptotic behavior.",
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Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. / Wu, Yen Lin; Chen, Zhi-You; Chern, Jann Long; Kabeya, Yoshitsugu.

In: Communications on Pure and Applied Analysis, Vol. 13, No. 2, 01.03.2014, p. 949-960.

Research output: Contribution to journalArticle

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AU - Kabeya, Yoshitsugu

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AB - In this article, we consider the following semilinear elliptic equation on the hyperbolic space ΔHnu - λu + |u|p-1u = 0 on Hn n\{Q} where ΔH n is the Laplace-Beltrami operator on the hyperbolic space Hn = {(x 1, · · · xn, xn+1)|x 1 2 + · · · + xn 2 - x n+1 2 = -1}, n > 10, p > 1, λ > 0, and Q = (0, · · · 0, 1). We provide the existence and uniqueness of a singular positive "radial" solution of the above equation for big p (greater than the Joseph-Lundgren exponent, which appears if n > 10) as well as its asymptotic behavior.

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