Existence of solutions for critical Henon equations in hyperbolic spaces

• Main Authors: Haiyang He, Jing Qiu
• Format: Article
• Language: English
• Published: Texas State University 2013-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Henon equations; mountain pass theorem; critical growth; hyperbolic space
• Online Access: http://ejde.math.txstate.edu/Volumes/2013/05/abstr.html

In this article, we use variational methods to prove that for a suitable value of $lambda$, the problem $$displaylines{ -Delta_{mathbb{B}^N}u=(d(x))^{alpha}|u|^{2^{*}-2}u+lambda u, quad ugeq 0,quad uin H_0^1(Omega') }$$ possesses at least one non-trivial solution u as $alphao 0^+$, 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 $mathbb{B}^N$, $Ngeq 4$, $2^*=2N/(N-2)$.