Normalized solutions for a fractional $N/s$-Laplacian Choquard equation with exponential critical nonlinearities

TL;DR

This paper establishes the existence of ground state solutions for a fractional N/s-Laplacian Choquard equation with exponential critical nonlinearities, using variational methods under prescribed norm constraints.

Contribution

It introduces a novel approach to solve a fractional Choquard equation with exponential critical growth, proving solutions exist for any prescribed norm.

Findings

Existence of ground state solutions for the equation.

Application of constraint variational method and minimax technique.

Solutions hold for any prescribed norm value.

Abstract

In this paper, we are concerned with the following fractional $N/s$-Laplacian Choquard equation \begin{align*} \begin{cases} (-\Delta)^s_{N/s}u=\lambda |u|^{\frac{N}{s}-2}u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}|u|^{N/s} \mathrm{d}x=a^{N/s}, \end{cases} \end{align*} where $s\in(0,1)$, $1<\frac{N}{s}\in \mathbb{N}^+$, $a>0$ is a prescribed constant, $\lambda\in \mathbb{R}$, $I_\mu(x)=\frac{1}{|x|^{\mu}}$ with $\mu\in(0,N)$, $F$ is the primitive function of $f$, and $f$ is a continuous function with exponential critical growth of Trudinger-Moser type. Under some suitable assumptions on $f$, we prove that the above problem admits a ground state solution for any given $a>0$, by using the constraint variational method and minimax technique.