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

TL;DR

This paper proves the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities, using topological methods to estimate the number of solutions based on the potential's properties.

Contribution

It introduces new results on the existence and multiplicity of solutions for fractional Choquard equations with mixed nonlinearities, employing Lusternik-Schnirelmann category theory.

Findings

Multiple normalized solutions are established for the equation.

The number of solutions is estimated by the category of the potential's minimum points.

The approach applies variational methods to fractional nonlocal equations.

Abstract

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N, $$ subject to the constraint $$ \int_{\mathbb{R}^N}|u|^2 \mathrm{d}x=a>0, $$ where $N>2 s, s \in(0,1), \alpha \in(0, N), \frac{N+\alpha}{N}<q<\frac{N+2 s+\alpha}{N}<p\leq \frac{N+\alpha}{N-2 s}$, $\epsilon>0$ is a parameter, and $\lambda \in \mathbb{R}$ 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 $V$.