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.