On mass - critical NLS with local and non-local nonlinearities

TL;DR

This paper studies a mass-critical nonlinear Schrödinger equation with local and non-local nonlinearities, proving ground state existence, non-degeneracy, and constructing minimal mass blowup solutions with specific parameters.

Contribution

It introduces a perturbation method to establish ground state non-degeneracy and constructs minimal mass blowup solutions parametrized by energy and momentum.

Findings

Existence of ground states $Q_$ with non-degeneracy.

Construction of minimal mass blowup solutions.

Ground state properties crucial for blowup analysis.

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+\Delta u+|u|^\frac{4}{3}u+\mu\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $\mu>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_{\mu}$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_{\mu}$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|_{L^2}=\|Q_{\mu}\|_{L^2}$. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy $E_{\mu}(u_0)>0$ and the momentum $P_{\mu}(u_0)$. In addition, the non-degeneracy property plays crucial role in this construction.