Normalized solutions to nonlinear Schr\"odinger equations with competing Hartree-type nonlinearities

TL;DR

This paper investigates normalized solutions to a nonlinear Schrödinger equation with competing Hartree nonlinearities, establishing existence results across different regimes and analyzing the well-posedness and dynamics of the related time-dependent problem.

Contribution

It provides the first comprehensive existence theory for ground states under various nonlinear regimes and explores the dynamics of solutions for the associated Cauchy problem.

Findings

Existence of ground states in subcritical, critical, and supercritical cases.

Analysis of well-posedness of the time-dependent equation.

Characterization of dynamical behaviors of solutions.

Abstract

In this paper, we consider solutions to the following nonlinear Schr\"odinger equation with competing Hartree-type nonlinearities, $$ -\Delta u + \lambda u=\left(|x|^{-\gamma_1} \ast |u|^2\right) u - \left(|x|^{-\gamma_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<\gamma_2 < \gamma_1 <\min\{N, 4\}$ and $\lambda \in \R$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.