New result for nonlinear Choquard equations: Doubly critical case

Abstract

In this paper, we consider the following equation: −Δu+u=(Iα∗F(u))F′(u)inRN,where N⩾5, α∈(0,N−4), and Iα is the Riesz potential, and F(u)≔12α♯|u|2α♯+12α∗|u|2α∗, where 2α♯=N+αN and 2α∗=N+αN−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).