Maximum decay rate for the nonlinear Schrodinger equation

In this paper, we consider global solutions for the following nonlinear Schrodinger equation iu(t) + Deltau + lambda\u\(alpha)u = 0, in R-N, with lambda is an element of R and 0 less than or equal to alpha < (4)/(N-2) (0 less than or equal to alpha < infinity if N = 1). We show that no nontrivial solution can decay faster than the solutions of the free Schrodinger equation, provided that u(0) lies in the weighted Sobolev space H-1(R-N) boolean AND L-2(\x\(2); dx), in the energy space, namely H-1(R-N), or in L-2(R-N), according to the different cases.