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

TL;DR

This paper proves the existence of solutions for a fractional Choquard-type equation on the real line involving the half-Laplacian, Riesz potential, and critical exponential growth, using variational methods and minimax estimates.

Contribution

It establishes the existence of solutions for a fractional Choquard equation with critical exponential growth, extending previous results to this specific nonlocal and nonlinear setting.

Findings

Existence of solutions proven using variational methods.

Solutions exist under critical exponential growth conditions.

Application of minimax estimates to fractional nonlocal equations.

Abstract

In this paper we study the following class of fractional Choquard--type equations \[ (-\Delta)^{1/2}u + u=\Big( I_\mu \ast F(u)\Big)f(u), \quad x\in\mathbb{R}, \] where $(-\Delta)^{1/2}$ denotes the $1/2$--Laplacian operator, $I_{\mu}$ is the Riesz potential with $0<\mu<1$ and $F$ is the primitive function of $f$. We use Variational Methods and minimax estimates to study the existence of solutions when $f$ has critical exponential growth in the sense of Trudinger--Moser inequality.