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

Abstract

We consider the following nonlinear Schrödinger equation with double power nonlinearities and an inverse power potential: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2σ}}u=0 \] in ${\mathbb{R}}^N$. From the classical argument, the solution with subcritical mass (${\Vert u_0\Vert }_2<{\Vert Q\Vert }_2$) 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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2007.15968; text overlap with arXiv:2108.06205