Uniqueness of positive multi-lump bound states of nonlinear Schrodinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schrodinger equation i (h) over bar partial derivativepsi/partial derivativet = -(h) over bar (2)Deltapsi + Vpsi - gamma\psi\(p-2)psi for sufficiently small (h) over bar > 0, where gamma > 0.2 < p < 2N/N-2 for N greater than or equal to 3 and 2 < p < +infinity for N = 1. 2. V is bounded on R-N. For any finite collection {a(1)..... a(k)} of nondegenerate critical points of V, we show the uniqueness of solutions of the form e(-iEt/(h) over bar) u(x) for E < inf(x is an element of RN) V(x), where u is positive on RN and is a small perturbation of a sum of one-lump solutions concentrated near a(1).....a(k). respectively for sufficiently small <(h)over bar> > 0.