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.