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

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 $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. 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 $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|_{L^2}=\|Q_μ\|_{L^2}$. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy $E_μ(u_0)>0$ and the momentum $P_μ(u_0)$. In addition, the non-degeneracy property plays crucial role in this construction.

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