The existence of solutions for Hénon equation in hyperbolic space

Abstract

In this paper, we use the variational methods to study the following problem \begin{equation} {-}Δ_{\mathbf{B}^{N}}u=(d(x))^{α}|u|^{p-2}u, u\in H_{r}^{1}(\mathbf{B}^{N}) \label{Lb1} \end{equation} in Hyperbolic space $\mathbf{B}^{N}$, where $\alpha>0$, $d(x)=d_{\mathbf{B}^{N}}(0,x)$, and $H_{r}^{1}(\mathbf{B}^{N})$ denote the Sobolev space of radial $H^{1}(\mathbf{B} ^{N})$ function on the disc model of the Hyperbolic space $\mathbf{B}^{N}$ and $\Delta_{\mathbf{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbf{B}^{N}$, $N\geq 3$. Unlike the corresponding problem in Euclidean space $\mathbf{R}^{N}$, we prove that there exists a positive solution of problem (1) provided that $p\in (2, \frac{2N+2\alpha}{N-2})$ which will be contrasted with a result due to Gidas and Spruck [6].