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

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.