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

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a fractional Choquard equation with mixed nonlinearities, establishing new results under specific parameter conditions and identifying ground state solutions for small parameter values.

Contribution

The study introduces new existence and multiplicity results for normalized solutions of fractional Choquard equations with mixed nonlinearities, including ground state solutions for small parameters.

Findings

Proved existence of normalized solutions under certain parameter ranges.

Established multiplicity of solutions for the fractional Choquard equation.

Identified conditions for the existence of ground state normalized solutions.

Abstract

In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-\Delta)^s u = \lambda u + \alpha \left( I_\mu * |u|^q \right) |u|^{q-2} u + \left( I_\mu * |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \\[4pt] \displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0. \end{array} \right. \] Here $N > 2s$, $s \in (0,1)$, $\mu \in (0, N)$, and the exponents satisfy \[ \frac{2N - \mu}{N} < q < p < \frac{2N - \mu}{N - 2s}, \] 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.