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

TL;DR

The paper develops a rigorous variational framework for normalized solutions to a fractional Choquard equation with mixed nonlocal nonlinearities under an $L^2$ constraint. By introducing a Pohozaev manifold and a fibering analysis along mass-preserving scalings, it proves the existence of two distinct normalized states in the mixed subcritical/supercritical regime and describes their qualitative properties, including positivity and radial symmetry. It also demonstrates convergence of these solutions to the autonomous problem as the coupling parameter $\alpha\to0$, and provides ground-state results in the radial setting, including the $L^2$-critical regime. Overall, the work advances understanding of how nonlocal Hartree-type interactions interact with mass constraints to yield multiple normalized states and their asymptotic behavior.

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.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Definition 2.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.1