Abstract:

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation
-△u=∫_R^N (|u(y)|^p)/(|x-y|^α)dy|u(x)|^p-2u(x) in R^N,
where N ≥ 3, 0< α < min{4, N}. Suppose that 2< p < (2N-α)/(N-2), we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2N-α)/(N-2), if i(u) < ∞, then we have ∫_R^N ∫_R^N (|u(x)|^p|u(y)|^p)/(|x-y|^α) dxdy < ∞ and ∫_R^N|▽u|² dx=∫_R^N ∫_R^N(|u(x)|^p|u(y)|^p)/(|x-y|^α dxdy.

Key words: Liouville type theorem, Morse index, Choquard equation