On the Choquard equation with double critical Sobolev exponents in ℝ N

Abstract

In this paper, we study the existence and nonexistence results to the Choquard equation − Δ u = ( ∫ R N | u ( y ) | 2 α ∗ | x − y | α d y ) | u | 2 α ∗ − 2 u ± | u | q − 2 u in R N , where 2 α ∗ = 2 N − α N − 2 , 0 < α < N , 1 < q ≤ 2 ∗ , 2 ∗ = 2 N N − 2 , N ≥ 3 . We first use the Pohozaev-type identity to show the nonexistence of solutions for 1 < q < 2 ∗ . When the equation has double critical exponents, i.e. q = 2 ∗ , we obtain the existence of radial ground state solutions by the Nehari manifold and Mountain pass theorem.