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

TL;DR

This paper investigates the existence and behavior of minimal mass blow-up solutions for a nonlinear Schrödinger equation with a singular inverse potential, focusing on the critical mass threshold where solutions may blow up.

Contribution

It introduces new results on blow-up solutions at the critical mass for Schrödinger equations with a singular potential involving a logarithmic term.

Findings

Existence of minimal mass blow-up solutions identified.

Behavior of solutions at the critical mass threshold analyzed.

Conditions under which blow-up occurs are characterized.

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}\log|x|u=0 \] in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\|u\|_2<\|Q\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$. 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 ($\left\|u_0\right\|_2=\left\|Q\right\|_2$).