Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency

Abstract

We are concerned with the fractional Schrödinger equation ϵ2α(−Δ)αu+V(x)u=|u|2α∗−2u,x∈RN,where ϵ>0 is a parameter, 0<α<1, N≥3, 2α∗=2NN−2α, V∈LN2α(RN) is a nonnegative function and V is assumed to be zero in some region of RN, 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.