Remarks on minimal mass blow up solutions for a double power nonlinear Schr\"{o}dinger equation

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearity \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+|u|^{p-1}u=0,\quad 1<p<1+\frac{4}{N} \] in $\mathbb{R}^N$. For $N=1,2,3$, Le Coz-Martel-Rapha\"{e}l (2016) construct a minimal-mass blow-up solution. Moreover, the previous study derives blow-up rate of the blow-up solution. In this paper, we extend this result to the general dimension. Furthermore, we investigate the behaviour of the critical mass blow-up solution near the blow-up time.