This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ ps}(-\Delta)_{ p}^{ s}u+\varepsilon^{ qs}(-\Delta)_q^su+ Z(x)(|u|^{ p-2}u+|u|^{ q-2}u)=\varepsilon^{ \mu-N}[|x|^{ -\mu}*F(u)]f(u) \ \ \mbox{ in} \ \ \mathbb{ R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{ N}{ s}<q,$ and $0<\mu<N.$ 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 $\varepsilon>0$ small enough. In a certain sense, we generalize some previously known results.
Categories: math.AP