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Blowup results and concentration in focusing Schrödinger-Hartree equation

  • * Corresponding author: Liquan Mei

    * Corresponding author: Liquan Mei

The work is partially supported by Science Challenge Project, No. TZ2016002

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  • This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in $ \dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N) $ and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded $ \dot{H}^{S_c} $ norm definitely have concentration properties related to a fixed $ \dot{H}^{S_c} $ norm of certain standing waves.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 74H35.


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  • Figure 1.  Comparison of energy-mass

    Figure 2.  Comparison of $ H^1 $ norm

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