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

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

doi:10.3934/cpaa.2014.13.949

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

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

Department of Mathematics

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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 Δ_H ⁿ is the Laplace-Beltrami operator on the hyperbolic space Hⁿ = {(x ₁, · · · x_n, x_n+1)|x ₁ ² + · · · + x_n 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 language	English
Pages (from-to)	949-960
Number of pages	12
Journal	Communications on Pure and Applied Analysis
Volume	13
Issue number	2
DOIs	https://doi.org/10.3934/cpaa.2014.13.949
Publication status	Published - 2014 Mar 1

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All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Cite this

Wu, Y. L., Chen, Z-Y., Chern, J. L., & Kabeya, Y. (2014). Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure and Applied Analysis, 13(2), 949-960. https://doi.org/10.3934/cpaa.2014.13.949

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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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N2 - 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.

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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10.3934/cpaa.2014.13.949