Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $${\epsilon }^{ps}(-\mathrm{\Delta }{)}_p^{s}u+{\epsilon }^{qs}(-\mathrm{\Delta }{)}_q^{s}u+Z(x)(\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u)={\epsilon }^{\mu -N}[\mathrm{\mid }x{\mathrm{\mid }}^{-\mu }\ast F(u)]f(u)in{\mathbb{R}}^N,$$ where $s\in (0,1),$ $\epsilon >0$ is a parameter, $2\le p=\frac{N}{s}<q,$ and $0<\mu <N\mathrm{.}$ The nonlinear function $f$ has an exponential growth at infinity and the continuous potential function $Z$ satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for $\epsilon >0$ small enough. In a certain sense, we generalize some previously known results.