THE MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF-CHOQUARD EQUATION WITH MAGNETIC FIELDS

摘要

In this paper,we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields:(aε^(2s)+bε^(4 s-3)[u]_(ε)^(2),A/ε)(-Δ)_(A/ε)^(s)u+V(x)u=ε^(-α)(Iα*F(|u|^(2)))f(|u|^(2))u in R^(3).Hereε>0 is a small parameter,a,b>0 are constants,s E(0,1),(-Δ)As is the fractional magnetic Laplacian,A:R^(3)→R^(3) is a smooth magnetic potential,Iα=Γ(3-α/2)/2απ3/2Γ(α/2)·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 C^(1) 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∈R^(3):V(x)=inf V}by applying the penalization method and LjusternikSchnirelmann theory for the above equation.