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February 2013 The existence of solutions for Hénon equation in hyperbolic space
Haiyang He
Proc. Japan Acad. Ser. A Math. Sci. 89(2): 24-28 (February 2013). DOI: 10.3792/pjaa.89.24

Abstract

In this paper, we use the variational methods to study the following problem \begin{equation} {-}Δ_{\mathbf{B}^{N}}u=(d(x))^{α}|u|^{p-2}u, u\in H_{r}^{1}(\mathbf{B}^{N}) \label{Lb1} \end{equation} in Hyperbolic space $\mathbf{B}^{N}$, where $\alpha>0$, $d(x)=d_{\mathbf{B}^{N}}(0,x)$, and $H_{r}^{1}(\mathbf{B}^{N})$ denote the Sobolev space of radial $H^{1}(\mathbf{B} ^{N})$ function on the disc model of the Hyperbolic space $\mathbf{B}^{N}$ and $\Delta_{\mathbf{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbf{B}^{N}$, $N\geq 3$. Unlike the corresponding problem in Euclidean space $\mathbf{R}^{N}$, we prove that there exists a positive solution of problem (1) provided that $p\in (2, \frac{2N+2\alpha}{N-2})$ which will be contrasted with a result due to Gidas and Spruck [6].

Citation

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Haiyang He. "The existence of solutions for Hénon equation in hyperbolic space." Proc. Japan Acad. Ser. A Math. Sci. 89 (2) 24 - 28, February 2013. https://doi.org/10.3792/pjaa.89.24

Information

Published: February 2013
First available in Project Euclid: 30 January 2013

zbMATH: 1268.58019
MathSciNet: MR3024271
Digital Object Identifier: 10.3792/pjaa.89.24

Subjects:
Primary: 35J60 , 58J05

Keywords: Hénon equation , Hyperbolic space , hyperbolic symmetry solution , Mountain pass theorem

Rights: Copyright © 2013 The Japan Academy

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Vol.89 • No. 2 • February 2013
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