A Class of Nonlinear Schrödinger Equations with Concentrated Nonlinearity

Abstract

We consider the nonlinear Schrödinger 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^ρ are given. We also prove the conservation of the L²-norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space H¹. Finally we prove the existence of blow-up solutions for strongly attractive interaction.