Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 208, pp. 1-13.

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

Authors: Haiyang He (Hunan Normal Univ., Changsha, China)
         Wei Wang   (Hunan Normal Univ., Changsha, China)

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.