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

Abstract

In this paper, we study the following class of fractional Choquard-type equations $$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$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.