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

We consider the following nonlinear Schrödinger equation with the double L-critical nonlinearities iut +∆u+ |u| 4 3u+ μ ( |x| ∗ |u| ) u = 0 in R, where μ > 0 is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state Qμ. 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 Qμ. We then show the existence of finite time blowup solution with minimal mass ‖u0‖L2 = ‖Qμ‖L2. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy Eμ(u0) > 0 and the momentum Pμ(u0). In addition, the nondegeneracy property plays crucial role in this construction. keywords: Nonlinear Schrödinger equation; Non-degeneracy; Mass-Critical; Blow-up; Local and non local nonlinearities

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