Multiplicity of semiclassical states for fractional Schrodinger equations with critical frequency

We are concerned with the fractional Schrodinger equation epsilon(2 alpha)(-Delta)(alpha)u + V(x)u = u vertical bar u vertical bar(2)*alpha(-2)u, x is an element of R-N, where epsilon > 0 is a parameter, 0 < alpha < 1, N >= 3, 2(alpha)* = 2N/N-2 alpha, V is an element of L-N/2 alpha(R-N) is a nonnegative function and V is assumed to be zero in some region of R-N, which means it is of the critical frequency case. By virtue of a global compactness lemma, two barycenter functions and Lusternik-Schnirelman theory, we show the multiplicity of high energy semiclassical states. (C) 2019 Elsevier Ltd. All rights reserved.