Electron. J. Diff. Equ., Vol. 2013 (2013), No. 05, pp. 1-11.

Existence of solutions for critical Henon equations in hyperbolic spaces

Haiyang He, Jing Qiu

Abstract:
In this article, we use variational methods to prove that for a suitable value of $\lambda$, the problem
$$\displaylines{ -\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{2^{*}-2}u+\lambda u, \quad u\geq 0,\quad u\in H_0^1(\Omega') }$$
possesses at least one non-trivial solution u as $\alpha\to 0^+$, where $\Omega'$ is a bounded domain in Hyperbolic space $\mathbb{B}^N$, $d(x)=d_{\mathbb{B}^N}(0,x)$. $\Delta_{\mathbb{B}^N}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^N$, $N\geq 4$, $2^*=2N/(N-2)$.

Submitted June 13, 2012. Published January 8, 2013.
Math Subject Classifications: 58J05, 35J60.
Key Words: Henon equations; mountain pass theorem; critical growth; hyperbolic space.

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Haiyang He
College of Mathematics and Computer Science
Key Laboratory of High Performance Computing
and Stochastic Information Processing
Hunan Normal University, Changsha, Hunan 410081, China
email: hehy917@yahoo.com.cn
Jing Qiu
College of Mathematics and Computer Science
Key Laboratory of High Performance Computing
and Stochastic Information Processing
Hunan Normal University, Changsha, Hunan 410081, China
email: qiujing0626@163.com

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