The nonlinear Schr\"odinger equations with combined nonlinearities of power-type and Hartree

TL;DR

This paper provides a comprehensive analysis of nonlinear Schrödinger equations with combined power-type and Hartree nonlinearities across dimensions, using Morawetz estimates and perturbation methods to understand their interactions in energy space.

Contribution

It offers a nearly complete characterization of the interactions between these nonlinearities in the energy space for dimensions n≥3, under certain structural conditions.

Findings

Detailed understanding of nonlinear interactions

Conditions for well-posedness and scattering

Framework applicable to various dimensions

Abstract

This paper is devoted to a comprehensive study of the nonlinear Schr\"odinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\ge3. With some structural conditions, a nearly whole picture of the interactions of these nonlinearities in the energy space is given. The method is based on the Morawetz estimates and perturbation principles.