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

Abstract

We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\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 $\Vert u_0{\Vert }_{L^2}=\Vert Q_{\mu }{\Vert }_{L^2}$. 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.

39pages. arXiv admin note: text overlap with arXiv:1001.1627, arXiv:1203.2476 by other authors