Minimal mass blow-up solutions for double power nonlinear Schr\"{o}dinger equations with an inverse power potential

TL;DR

This paper studies the existence and behavior of minimal-mass blow-up solutions for a nonlinear Schrödinger equation with double power nonlinearities and an inverse power potential, especially when the effects of the nonlinearity and potential cancel each other.

Contribution

It investigates the existence and properties of minimal-mass blow-up solutions in cases where the nonlinear term and potential have opposing signs, extending previous results.

Findings

Existence of minimal-mass blow-up solutions when effects cancel

Lower bounds for finite-time blow-up solutions with critical mass

Positivity of energies for certain parameter regimes

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearities and an inverse power potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2\sigma}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\left\|u_0\right\|_2<\left\|Q\right\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$, where $Q$ is the ground state of the mass-critical problem. Previous results show the existence of a minimal-mass blow-up solution for the equation with $C_1>0$ and $C_2=0$ or $C_1=0$ and $C_2>0$ and investigate the behaviour of the solution near the blow-up time. Moreover, they have suggested that a subcritical power nonlinearity and an inverse power potential behave in a similar way with respect to blow-up. On the other hand, the previous results also show the nonexistence of a minimal-mass blow-up solution for the equation with $C_1<0$ and $C_2=0$ or $C_1=0$ and $C_2<0$. In this paper, we investigate the existence and behaviour of a minimal-mass blow-up solution for the equation with $C_1>0>C_2$ or $C_1<0<C_2$, that is the subcritical power nonlinearity and the inverse power potential cancel each other's effects. Furthermore, we give a lower estimate of the arbitrary finite-time blow-up solution with critical mass and show that the energies of critical-mass blow-up solutions are positive when $(C_1,C_2,p,\sigma)$ is under certain conditions.