Existence of solutions for critical Henon equations in hyperbolic spaces

Abstract

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

Keywords

• Henon equations
• mountain pass theorem
• critical growth
• hyperbolic space