In this paper, we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields:
(aε2s+bε4s−3[u]2ε,A/ε)(−Δ)sA/εu+V(x)u=ε−α(Iα∗F(|u|2))f(|u|2)u in R3.
Here
ε>0 is a small parameter,
a,b>0 are constants,
s∈(0 is the fractional magnetic Laplacian,
A:R3→R3 is a smooth magnetic potential,
Iα=Γ(3−α2)2απ32Γ(α2)⋅1|x|α is the Riesz potential, the potential
V is a positive continuous function having a local minimum, and
f:R→R is a
C1 subcritical nonlinearity. Under some proper assumptions regarding
V and
f, we show the multiplicity and concentration of positive solutions with the topology of the set
M:={x∈R3:V(x)=infV} by applying the penalization method and Ljusternik-Schnirelmann theory for the above equation.