MULTIPLE CONCENTRATING SOLUTIONS FOR A FRACTIONAL KIRCHHOFF EQUATION WITH MAGNETIC FIELDS

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: (alpha epsilon(2s) + b epsilon(4s-3)[u](A/epsilon)(2)) (-Delta)(A/epsilon)(s) u + V(x) u = f(vertical bar u vertical bar(2))u in R-3, where epsilon > 0 is a small parameter, a, b > 0 are constants, s is an element of (3/4, 1), (-Delta)(A)(s) is the fractional magnetic Laplacian, A : R-3 -> R-3 is a smooth magnetic potential, V : R-3 -> R is a positive continuous electric potential satisfying local conditions and f : R -> R is a C-1 subcritical nonlinearity. Applying penalization techniques, fractional Kato's type inequality and Ljusternik-Schnirelmann theory, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.