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Negative Energy Ground States for the L 2-Critical NLSE on Metric Graphs

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Abstract

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with L 2-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo–Nirenberg inequalities and by estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

    Riccardo Adami, Enrico Serra & Paolo Tilli

Authors
  1. Riccardo Adami
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  2. Enrico Serra
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  3. Paolo Tilli
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Correspondence to Riccardo Adami.

Additional information

Communicated by W. Schlag

Riccardo Adami is partially supported by the FIRB 2012 project “Dispersive dynamics: Fourier Analysis and Variational Methods”.

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Adami, R., Serra, E. & Tilli, P. Negative Energy Ground States for the L 2-Critical NLSE on Metric Graphs. Commun. Math. Phys. 352, 387–406 (2017). https://doi.org/10.1007/s00220-016-2797-2

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  • DOI: https://doi.org/10.1007/s00220-016-2797-2

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