We study the existence of normalized solutions to the following Choquard equation with $F$ being a Berestycki-Lions type function \begin{ equation*} \begin{ cases} -\Delta u+\lambda u=(I_{ \alpha}\ast F(u))f(u),\quad \text{ in}\ \mathbb{ R}^N, \\ \int_{ \mathbb{ R}^N}|u|^2dx=\rho^2, \end{ cases} \end{ equation*} where $N\geq 3$, $\rho>0$ 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.
Categories: math.AP