Multiplicity and Concentration Results for Some Fractional Double Phase Choquard Equation with Exponential Growth

In this paper, we study the fractional Choquard equation in R N ε ps ( − Δ ) p s u + ε qs ( − Δ ) q s u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = ε μ − N [ | x | − μ * F ( u ) ] f ( u ) , where ε is a positive parameter, N = p s , 2 ≤ p < q , s ∈ ( 0 , 1 ) , 0 < μ < N . The nonlinear function f has an exponential growth at infinity and the potential function V is continuous in R N and satisfies suitable natural conditions. Using the Ljusternik–Schnirelmann category theory and variational methods, we establish multiplicity and concentration of positive solutions for small values of the parameter ε > 0 .