On the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

TL;DR

This work establishes the existence of multiple normalized solutions for a nonlinear fractional Choquard equation with mixed nonlinearities and a nonautonomous potential $V(\varepsilon x)$. By formulating a constrained variational problem on the $L^2$-sphere and employing a truncated energy functional, the authors derive PS-type compactness and concentration results. They relate the number of normalized solutions to the topology of the zero-set $\mathcal{M}=\{x: V(x)=0\}$ via Lusternik–Schnirelmann category, proving at least $\mathrm{cat}_{\mathcal{M}_\delta}(\mathcal{M})$ pairs of solutions for small $\varepsilon$, with negative Lagrange multipliers and negative energy. The analysis combines nonlocal Choquard terms, Hardy–Littlewood–Sobolev inequalities, and concentration-compactness type arguments to capture localization near the minimum set of the potential, yielding a robust multiplicity result in the fractional setting.

Abstract

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(εx)u=λu+\left(I_α*|u|^q\right)|u|^{q-2} u+\left(I_α*|u|^p\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N, $$ subject to the constraint $$ \int_{\mathbb{R}^N}|u|^2 \mathrm{d}x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N), \frac{N+α}{N}<q<\frac{N+2 s+α}{N}<p\leq \frac{N+α}{N-2 s}$, $ε>0$ is a parameter, and $λ\in \mathbb{R}$ serves as an unknown parameter acting as a Lagrange multiplier. By employing the Lusternik-Schnirelmann category theory, we estimate the number of normalized solutions to this problem by virtue of the category of the set of minimum points of the potential function $V$.

Theorems & Definitions (22)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6