Existence of normalized solutions to Choquard equation with general mixed nonlinearities

Abstract

We study the existence of normalized solutions to the following Choquard equation with $F$ being a Berestycki-Lions type function \begin{equation*} \begin{cases} -Δu+λu=(I_α\ast F(u))f(u),\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=ρ^2, \end{cases} \end{equation*} where $N\ge 3$, $\rho >0$ is assigned, $\alpha \in (0,N)$, $I_{\alpha }$ is the Riesz potential, and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. Here, the general nonlinearity $F$ contains the $L^2$-subcritical and $L^2$-supercritical mixed case, the Hardy-Littlewood-Sobolev lower critical and upper critical cases.