The dynamics of the 3D radial NLS with the combined terms

Abstract

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms iu_t + Δu = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1({\mathbb{R}}^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $i{u}_t+\Delta u=-|u{|}^{4}u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, ${\dot{H}}^1$-subcritical perturbation $|u{|}^{2}u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.

46pages

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