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Stable standing waves of nonlinear fractional Schrödinger equations

  • * Corresponding author: Qidi Zhang

    * Corresponding author: Qidi Zhang 

Z. Li is supported by Natural Science Foundation of Hebei Province, No. A2022205007, by Science and Technology Project of Hebei Education Department, No. QN2022047, and by Science Foundation of Hebei Normal University, No. L2021B05. Z. Zhang is supported by National Natural Science Foundation of China, No. 12031015, 11771428, 12026217

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  • We study the existence and orbital stability of standing waves of nonlinear fractional Schrödinger equations with a general nonlinear term

    $ \begin{equation*} \mathrm{i} u_t-\left(-\Delta\right)^s u +f\left(u\right) = 0, \ \left(t, x\right)\in\mathbb{R}_+\times\mathbb{R}^N. \end{equation*} $

    We investigate the minimizing problem with $ L^2 $-constraint:

    $ \begin{equation*} E_{\alpha} = \inf\Big\{\frac{1}{2}\int_{\mathbb{R}^N}\!|(-\Delta)^{\frac{s}{2}}u|^2\mathrm{d}x-\int_{\mathbb{R}^N}\!F(|u|)\mathrm{d}x\ \Big|\ u\in H^{s}(\mathbb{R}^N), \|u\|^2_{L^2(\mathbb{R}^N)} = \alpha\Big\}. \end{equation*} $

    The existence and non-existence of global minimizers with respect to $ E_{\alpha} $ are established for all possible values of $ \alpha. $ Under some general assumptions on the nonlinear term $ f(u) $, there exists a constant $ \alpha_0\ge 0 $ such that a global minimizer exists for $ E_\alpha $ for all $ \alpha>\alpha_0 $, and there is no global minimizer with respect to $ E_{\alpha} $ for all $ 0<\alpha<\alpha_0. $ By virtue of concentration-compactness argument and the strict subadditivity of $ E_\alpha $, the strong convergence of minimizing sequence is obtained. Moreover, we present some criteria which determine $ \alpha_0 = 0 $ or $ \alpha_0>0 $, and the existence of global minimizers for $ E_{\alpha_0}. $ Besides, we show the orbital stability of the global minimizers set. Finally, we prove that an energy minimizer is a least action solution by Pohozaev identity.

    Mathematics Subject Classification: Primary: 35R11, 35A01; Secondary: 35A15.


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