Sharp Exponent of Stable Standing Waves for the Perturbated Hartree Equation

TL;DR

This paper investigates the stability and blowup criteria of standing waves in a perturbed mass-critical Hartree equation, employing variational methods and profile decomposition to establish sharp stability thresholds.

Contribution

It introduces improved variational techniques and sharp stability criteria for standing waves in the perturbed Hartree equation, extending previous methods by Zhang (2005).

Findings

Sharp blowup criteria established

Orbital stability conditions derived

Strong instability results obtained

Abstract

This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital stability, and strong instability of standing waves without any frequency constraint are obtained. This improves the cross constrained variational argument proposed by Zhang (2005).