Minimal mass blow-up solutions for nonlinear Schr\"{o}dinger equations with a potential

Abstract

We consider a mass critical nonlinear Schr\"{o}dinger equation with a real-valued potential. In this work, we construct a minimal mass solution that blows up at finite time, under weaker assumptions on spatial dimensions and potentials than Banica, Carles, and Duyckaerts (2011). Moreover, we show that the blow-up solution converges to a blow-up profile. Furthermore, we improve some parts of the arguments in Rapha\"{e}l and Szeftel (2011) and Le Coz, Martel, and Rapha\"{e}l (2016).