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

TL;DR

This paper studies the existence and behavior of minimal mass blow-up solutions for a nonlinear Schrödinger equation with Hartree nonlinearity, extending previous work on inverse power potentials.

Contribution

It introduces analysis of minimal mass blow-up solutions specifically for Schrödinger equations with Hartree nonlinearity, a scaling-similar nonlinear term.

Findings

Existence of minimal mass blow-up solutions established.

Behavior of solutions near blow-up characterized.

Comparison with inverse power potential cases provided.

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2\sigma}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are interested in the existence and behaviour of minimal mass blow-up solutions. Previous studies have shown the existence of minimal mass blow-up solutions with an inverse power potential and investigated the behaviour of the solution. In this paper, we investigate Hartree nonlinearity, which is a nonlinear term similar to the inverse power-type potential in terms of scaling.