NONLINEAR CHOQUARD EQUATIONS ON HYPERBOLIC SPACE

In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation -Delta(B)Nu = integral(BN)vertical bar u(y)(p)/vertical bar 2sinh rho(T-y(x))/2 vertical bar mu dV(y) . vertical bar u vertical bar(P-2)u + lambda u on the hyperbolic space B-N, where Delta(BN) denotes the Laplace-Beltrami operator on B-N, sinh rho(T-y(x))/2 = vertical bar T-y(x)vertical bar/root 1 - vertical bar T-y(x)vertical bar(2) = vertical bar x - y vertical bar/root(1 - vertical bar x vertical bar(2))(1 - vertical bar y vertical bar(2)), lambda is a real parameter, 0 < mu < N, 1 < p <= 2(mu)*, N >= 3 and 2(mu)* := 2N-mu/N-2 is the critical exponent in the sense of the Hardy Littlewood Sobolev inequality.