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

2022/01/12 by Georgiev, Vladimir, Li, Yuan
#35B40 #35B44 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2201.04500

Abstract

We consider the following nonlinear Schrödinger equation with the double L²-critical nonlinearities iu_t+Δu+|u|^(4)/(3)u+μ(|x|^-2*|u|²)u=0 in ℝ³, 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 ‖u₀‖_L²=‖Q_μ‖_L². More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy E_μ(u₀)>0 and the momentum P_μ(u₀). In addition, the non-degeneracy property plays crucial role in this construction.

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