TY  - JOUR
ID  - He2022
LB  - He2022
AU  - He, Haiyang
TI  - Nonlinear Choquard equations on hyperbolic space
ST  - Nonlinear Choquard equations on hyperbolic space
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2022
VL  - 42
IS  - 5
SP  - 691
EP  - 708
DO  - https://doi.org/10.7494/OpMath.2022.42.5.691
UR  - https://doi.org/10.7494/OpMath.2022.42.5.691
AB  - 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.
KW  - nonlinear Choquard equation
KW  - hyperbolic space
KW  - existence solutions
KW  - Hardy-Littlewood-Sobolev inequality
SN  - 1232-9274
ER  -