Nonlinear Choquard equations on hyperbolic space

Abstract

In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation $-\Delta_{\mathbb{B}^{N}}u=\int_{\mathbb{B}^N}\dfrac{|u(y)|^{p}}{|2\sinh\frac{\rho(T_y(x))}{2}|^\mu} dV_y \cdot |u|^{p-2}u +\lambda u$ on the hyperbolic space $\mathbb{B}^{N}$, where $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^{N}$, $\sinh\frac{\rho(T_y(x))}{2}=\dfrac{|T_y(x)|}{\sqrt{1-|T_y(x)|^2}}=\dfrac{|x-y|}{\sqrt{(1-|x|^2)(1-|y|^2)}},$ $\lambda$ is a real parameter, $0\lt \mu\lt N$, $1\lt p\leq 2_\mu^*$, $N \geq 3$ and $2_\mu^*:=\frac{2N-\mu}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.