Stable standing waves of nonlinear fractional Schrödinger equations

We study the existence and orbital stability of standing waves of nonlinear fractional Schrödinger equations with a general nonlinear term \begin{document}$ \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*} $\end{document} We investigate the minimizing problem with \begin{document}$ L^2 $\end{document} -constraint: \begin{document}$ \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*} $\end{document} The existence and non-existence of global minimizers with respect to \begin{document}$ E_{\alpha} $\end{document} are established for all possible values of \begin{document}$ \alpha. $\end{document} Under some general assumptions on the nonlinear term \begin{document}$ f(u) $\end{document} , there exists a constant \begin{document}$ \alpha_0\ge 0 $\end{document} such that a global minimizer exists for \begin{document}$ E_\alpha $\end{document} for all \begin{document}$ \alpha>\alpha_0 $\end{document} , and there is no global minimizer with respect to \begin{document}$ E_{\alpha} $\end{document} for all \begin{document}$ 0<\alpha<\alpha_0. $\end{document} By virtue of concentration-compactness argument and the strict subadditivity of \begin{document}$ E_\alpha $\end{document} , the strong convergence of minimizing sequence is obtained. Moreover, we present some criteria which determine \begin{document}$ \alpha_0 = 0 $\end{document} or \begin{document}$ \alpha_0>0 $\end{document} , and the existence of global minimizers for \begin{document}$ E_{\alpha_0}. $\end{document} 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.

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