2018
DOI: 10.3934/dcds.2018025
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Energy-critical NLS with potentials of quadratic growth

Abstract: Consider the global wellposedness problem for nonlinear Schrödinger equation2 |x| 2 was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials.We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.

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Cited by 7 publications
(3 citation statements)
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“…This paper is not the first to implement the credo that to treat critical dispersive equations with broken symmetries, one must first solve the limiting problems where the symmetries are restored. Previous examples include [15,16,17,18,19,24,26,27,31,36]. This is the first model to be studied, however, which retains scale invariance, but loses space translation invariance.…”
Section: Introductionmentioning
confidence: 99%
“…This paper is not the first to implement the credo that to treat critical dispersive equations with broken symmetries, one must first solve the limiting problems where the symmetries are restored. Previous examples include [15,16,17,18,19,24,26,27,31,36]. This is the first model to be studied, however, which retains scale invariance, but loses space translation invariance.…”
Section: Introductionmentioning
confidence: 99%
“…, the equation is energy-critical: the above conclusion is known to remain when V = 0 [31], when V is an isotropic quadratic potential [19], or when V is harmonic at infinity [17].…”
Section: Resultsmentioning
confidence: 99%
“…The global well-posedness of (1.1) has been established in the energy-subcritical case by R. Carles [14,16] in the defocusing case µ = 1, and by J. Zhang [93] in the focusing case µ = −1 when the initial energy is assumed to be less than the energy of the ground state of the related elliptic equation. The Cauchy problem for the equation (1.1) with quadratic potential (that is, ω = 1) in the energy-critical case was considered by R. Killip, M. Visan, and X. Zhang [67] in the radial case, and in the general case later by C. Jao [55,56]. They proved the global well-posedness for the defocusing case and also for the focusing case when the initial energy (resp.…”
mentioning
confidence: 98%