Blow-up solutions for the Schrodinger equation in dimension three with a concentrated nonlinearity

摘要

We present some results on the blow-up phenomenon for the Schr?dinger equation in dimension three with a nonlinear term supported in a fixed point. We find sufficient conditions for the blow-up exploiting the moment of inertia of the solution and the uncertainty principle. In the critical case, we discuss the additional symmetries of the equation and construct a family of explicit blow-up solutions.