Concentrating standing waves for the fractional nonlinear Schrodinger equation

Abstract

We consider the semilinear equation

epsilon(2s)(-Delta)(s)u + V(x)u - u(p) = 0, u > 0, u is an element of H-2s(R-N)

where 0 < s < 1, 1 < p < N+2s/N-2s, V (x) is a sufficiently smooth potential with inf(R) V(x) > 0, and epsilon > 0 is a small number. Letting w(lambda) be the radial ground state of (-Delta)(s) w(lambda) + lambda w(lambda) - w(lambda)(p) = 0 in H-2s (R-N), we build solutions of the form

u epsilon(x) similar to (k)Sigma(i=1)w lambda(i)((x - xi(epsilon)(i))/epsilon),

where lambda(i) = V(xi(epsilon)(i)) and the xi(epsilon)(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 and Weinstein [13] and Oh [24,25]. (C) 2013 Elsevier Inc. All rights reserved.