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Ground states of nonlinear Schrödinger equations with fractional Laplacians

  • * Corresponding author: Qinqin Zhang

    * Corresponding author: Qinqin Zhang

This work was supported by National Natural Science Foundation (11701114, 11471085) and the program for Changjiang scholars and Innovative Recearch Team in univesity (Grant No.IRT1226)

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  • Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians

    $ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u+u& = f(u), x\in\mathbb{R}^N,\\ u(x)&\geq 0. \end{aligned} \right. \end{equation*} $

    As an application, we prove the existence of symmetric ground states in the fractional Sobolev space $ H^\alpha (\mathbb{R}^N) $. These results improve some known ones in the literature. An example is also given to illustrate our results.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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