On the fractional Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

In this paper, we consider the fractional Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity {epsilon M-2s([u](s,Ag)(2))(-Delta)(ag)(s) u + V(x)u = vertical bar u vertical bar(2s*-2) u+ h(x, vertical bar u vertical bar(2))u, x is an element of R-N u(x) -> 0, as vertical bar x vertical bar -> infinity, where (-Delta)(s)(A epsilon) is the fractional magnetic operator with 0 < s < 1, 2(s)(*) = 2N/(N - 2s),M : R-0(+) > R+ is a continuous nondecreasing function, V : R-N -> R-0(+) and A : R-N -> R-N are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that epsilon < epsilon; and (ii) for any m* is an element of N, has m* pairs of solutions if epsilon < epsilon(m*), where epsilon and epsilon(m*). are sufficiently small positive numbers. Moreover, these solutions u(epsilon) -> 0 as epsilon -> 0. (C) 2017 Elsevier Ltd. All rights reserved.