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

Abstract

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\epsilon >0$ is a parameter, $2\le p=\frac{N}{s}<q,$ and $0<\mu <N.$ 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.