Threshold solutions for the nonlinear Schrödinger equation
نویسندگان
چکیده
We study the focusing NLS equation in $\mathbb{R}^N$ mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at mass-energy threshold $\mathcal{ME}(u\_0)=\mathcal{ME}(Q)$, where $Q$ is ground state. Previously, Duyckaerts–Merle studied behavior of solutions $H^1$-critical case, dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In Duyckaerts–Roudenko problem 3d cubic equation. this paper, we generalize results any dimension power nonlinearity entire range. show existence special solutions, $Q^\pm$, besides standing wave $e^{it}Q$, which exponentially approach positive time direction, but differ its negative time. classify level, showing either blow-up occurs finite (positive negative) time, or scattering both directions, solution equal to one three above, up symmetries. Our proof extends thus, giving an alternative result unifying critical cases. These are obtained studying linearized around some tailored approximate establish important decay properties functions associated spectrum Schrödinger operator, which, combination modulational stability coercivity operator on subspaces, allows us use a fixed-point argument solutions. Finally, prove uniqueness decaying sequence equations.
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ژورنال
عنوان ژورنال: Revista Matematica Iberoamericana
سال: 2022
ISSN: ['2235-0616', '0213-2230']
DOI: https://doi.org/10.4171/rmi/1337