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 $N>2s$, $s\in (0,1)$, $\mu \in (0,N)$, and the exponents satisfy \[ \frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s}, \] while $\alpha >0$ is a sufficiently small parameter, $\lambda \in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_{\mu }$ 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 $\alpha$ in a suitable range.

arXiv comment49 pages, comments are welcome