New result for nonlinear Choquard equations: Doubly critical case

In this paper, we consider the following equation: -Delta u + u = (I-alpha * F(u))F'(u) in R-N, where N >= 5, alpha is an element of (0, N - 4), and I-alpha is the Riesz potential, and F(u) :=1/2(alpha)(#) vertical bar u vertical bar(2 alpha#) + 1/2(alpha)(#)vertical bar u vertical bar(2 alpha#), where 2(alpha)(#) = N+alpha/N and 2(alpha)(*) = N+alpha/N-2 are lower and upper critical exponents in the sense of the Hardy-Littlewood-Sobolev inequality. By using the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative solution for above equation. Our result extends the result in Seok (2018). (C) 2019 Elsevier Ltd. All rights reserved.