Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 05, pp. 1-11.

Title: Existence of solutions for critical Henon equations in hyperbolic spaces
        
Authors: Haiyang He (Hunan Normal Univ., Changsha, China)
         Jing Qiu   (Hunan Normal Univ., Changsha, China)

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 08, 2013.
Math Subject Classifications: 58J05, 35J60.
Key Words: Henon equations; mountain pass theorem;
           critical growth; hyperbolic space.