On the existence of normalized solutions to a class of fractional Choquard equation with potentials

TL;DR

This paper proves the existence of normalized solutions for a nonlinear fractional Choquard equation with potentials using variational methods under specific conditions.

Contribution

It introduces new existence results for normalized solutions to a class of fractional Choquard equations with potentials, considering the Lagrange multiplier as an unknown.

Findings

Established existence of solutions under certain potential conditions.

Applied variational methods to fractional Choquard equations.

Analyzed the role of the Lagrange multiplier in normalization.

Abstract

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha *\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), \alpha \in(0, N)$, and $\frac{N+\alpha}{N} \leq q<p \leq \frac{N+\alpha+2 s}{N}$. Here, the parameter $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x), f(x)$, and $g(x)$, we establish several existence results for normalized solutions.