Minimal-mass blow-up solutions for nonlinear Schrödinger equations with an inverse potential

Abstract

We consider the following nonlinear Schrödinger equation with an inverse potential: i∂u∂t+Δu+|u|4Nu±1|x|2σu=0in RN. From the classical argument, the solution with subcritical mass (‖u‖2<‖Q‖2) is global and bounded in H1(RN). Here, Q is the ground state of the mass-critical problem. Therefore, we are interested in the existence and behaviour of blow-up solutions for the threshold (‖u0‖2=‖Q‖2). Previous studies investigate the existence and behaviour of the critical-mass blow-up solution when the potential is smooth or unbounded but algebraically tractable. There exist no results when classical methods cannot be used, such as the inverse power type potential. However, in this paper, we construct a critical-mass finite-time blow-up solution. Moreover, we show that the blow-up solution converges to a certain blow-up profile in the virial space.