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.

Show me the PDF file (276 KB), TEX file, and other files for this article.

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

Return to the EJDE web page