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

Abstract

In this paper, we study the following fractional Choquard-type equation with prescribed mass \begin{align*} \begin{cases} (-Δ)^{1/2}u=λu +(I_μ*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.

arXiv admin note: text overlap with arXiv:2211.13701