Concentration phenomena for a class of fractional Kirchhoff equations in RN with general nonlinearities

In this paper we study the following class of fractional Kirchhoff problems: ε2sM(ε2s−N[u]s2)(−Δ)su+V(x)u=f(u) in RN,u∈Hs(RN),u>0 in RN,where ε>0 is a small parameter, s∈(0,1), N≥2, (−Δ)s is the fractional Laplacian, V:RN→R is a positive continuous function, M:[0,∞)→R is a Kirchhoff function satisfying suitable conditions and f:R→R fulfills Berestycki–Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions (uε) which concentrates at a local minimum of V as ε→0.