Concentrating Standing Waves for the Fractional Nonlinear Schrödinger Equation

We consider the semilinear equation ε(−∆)u+ V (x)u− u = 0, u > 0, u ∈ H(R ) where 0 < s < 1, 1 < p < N+2s N−2s , V (x) is a sufficiently smooth potential with infR V (x) > 0, and ε > 0 is a small number. Letting wλ be the radial ground state of (−∆)wλ +λwλ−w λ = 0 in H 2s(RN ), we build solutions of the form uε(x) ∼ k ∑ i=1 wλi ((x− ξ ε i )/ε), where λi = V (ξ ε i ) and the ξ ε i approach suitable critical points of V . Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case s = 1. In particular such a solution exists around k nondegenerate critical points of V . For s = 1 this corresponds to the classical results by Floer-Weinstein [13] and Oh [21, 22].