Electron. J. Diff. Equ., Vol. 2013 (2013), No. 208, pp. 1-13.

Existence and asymptotic behavior of solutions for Henon equations in hyperbolic spaces

Haiyang He, Wei Wang

Abstract:
In this article, we consider the existence and asymptotic behavior of solutions for the Henon equation
$$\displaylines{
 -\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{p-2}u, \quad  x\in   \Omega\cr
  u=0  \quad   x\in \partial \Omega,
 }$$
where $\Delta_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space $\mathbb{B}^N$, $d(x)=d_{\mathbb{B}^N}(0,x)$, $\Omega \subset \mathbb{B}^N$ is geodesic ball with radius $1$, $\alpha>0, N\geq 3$. We study the existence of hyperbolic symmetric solutions when $2<p<\frac{2N+2\alpha}{N-2}$. We also investigate asymptotic behavior of the ground state solution when p tends to the critical exponent $2^* =\frac{2N}{N-2}$ with $N\geq 3$.

Submitted May 29, 2013. Published September 19, 2013.
Math Subject Classifications: 35J20, 35J60.
Key Words: Henon equation; hyperbolic space; asymptotic behavior; blow up.

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Haiyang He
College of Mathematics and Computer Science
Key Laboratory of High Performance Computing and
Stochastic Information Processing, Ministry of Education
Hunan Normal University, Changsha, Hunan 410081, China
email: hehy917@hotmail.com
Wei Wang
College of Mathematics and Computer Science
Key Laboratory of High Performance Computing and
Stochastic Information Processing, Ministry of Education
Hunan Normal University, Changsha, Hunan 410081, China
email: ww224182255@sina.com

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