Normalized solutions for a fractional Choquard-type equation with exponential critical growth in $\mathbb{R}$

TL;DR

This paper establishes the existence of normalized ground state solutions for a fractional Choquard-type equation with exponential critical growth in one dimension, using a minimax approach and variational methods.

Contribution

It introduces a novel variational framework to find normalized solutions for a fractional Choquard equation with exponential critical growth, extending previous results to this specific nonlocal setting.

Findings

Existence of at least one normalized ground state solution.

Application of minimax principle based on homotopy stable family.

Extension of variational methods to fractional Choquard equations with exponential growth.

Abstract

In this paper, we study the following fractional Choquard-type equation with prescribed mass \begin{align*} \begin{cases} (-\Delta)^{1/2}u=\lambda u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2 \mathrm{d}x=a^2, \end{cases} \end{align*} where $(-\Delta)^{1/2}$ denotes the $1/2$-Laplacian operator, $a>0$, $\lambda\in \mathbb{R}$, $I_\mu(x)=\frac{{1}}{{|x|^\mu}}$ with $\mu\in(0,1)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Trudinger-Moser inequality. By using a minimax principle based on the homotopy stable family, we obtain that there is at least one normalized ground state solution to the above equation.