A class of nonlinear Schrodinger equations with concentrated nonlinearity

We consider the nonlinear Schrodinger equation in dimension one with a nonlinearity concentrated in a finite number of points. Detailed results on the local existence of the solution in Fractional Sobolev spaces H-p are given. We also prove the conservation of the L-2-norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space H-1. Finally we prove the existence of blow-up solutions for strongly attractive interaction. (C) 2001 Academic Press.