Nonlinear Choquard equations: doubly critical case

Abstract

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_\alpha$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \frac{N}{N+\alpha}|u|^{\frac{N+\alpha}{N}}+\frac{N-2}{N+\alpha}|u|^{\frac{N+\alpha}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \geq 5$ and $\alpha + 4 < N$.