Concentrating standing waves for the fractional nonlinear Schrödinger equation

We consider the semilinear equationε2s(−Δ)su+V(x)u−up=0,u>0,u∈H2s(RN) where 00, and ε>0ε>0 is a small number. Letting wλwλ be the radial ground state of (−Δ)swλ+λwλ−wλp=0 in H2s(RN)H2s(RN), we build solutions of the formuε(x)∼∑i=1kwλi((x−ξiε)/ε), where λi=V(ξiε) and the ξiε approach suitable critical points of V  . Via a Lyapunov–Schmidt variational reduction, we recover various existence results already known for the case s=1s=1. In particular such a solution exists around k nondegenerate critical points of V  . For s=1s=1 this corresponds to the classical results by Floer and Weinstein [13] and Oh [24] and [25].