Nonlinear Choquard equations: Doubly critical case

Abstract

Consider nonlinear Choquard equations − Δ u + u = ( I α ∗ F ( u ) ) F ′ ( u ) in R N , lim x → ∞ u ( x ) = 0 , where I α denotes Riesz potential and α ∈ ( 0 , N ) . In this paper, we show that when F is doubly critical, i.e. F ( u ) = N N + α | u | N + α N + N − 2 N + α | u | N + α N − 2 , the nonlinear Choquard equation admits a nontrivial solution if N ≥ 5 and α + 4 < N .