Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation -Δu=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x)inRN,where N ≥ 3, 0 < α < {4,N}. Suppose that 2 < p < 2N-αN-2, we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=2N-αN-2, if i(u) < ∞ then we have ∫RN∫RN|u(x)|p|u(y)|p|x-y|αdxdy<∞ and ∫RN|∇u|2dx=∫RN∫RN|u(x)|p|u(y)|p|x-y|αdxdy.