Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

Abstract: In this paper we study the following fractional Choquard equation with mixed nonlinearities: [ \left{ \begin{array}{l} (-Δ)^s u = λu + α\left( I_μ* |u|^q \right) |u|^{q-2} u + \left( I_μ* |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \[4pt] \displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0. \end{array} \right. ] Here $N > 2s$, $s \in (0,1)$, $μ\in (0, N)$, and the exponents satisfy [ \frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s}, ] while $α> 0$ is a sufficiently small parameter, $λ\in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_μ$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $α$ in a suitable range.