Stable standing waves of nonlinear fractional Schrödinger equations

<p style='text-indent:20px;'>We study the existence and orbital stability of standing waves of nonlinear fractional Schrödinger equations with a general nonlinear term <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>We investigate the minimizing problem with <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-constraint: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>The existence and non-existence of global minimizers with respect to <inline-formula><tex-math id="M2">\begin{document}$ E_{\alpha} $\end{document}</tex-math></inline-formula> are established for all possible values of <inline-formula><tex-math id="M3">\begin{document}$ \alpha. $\end{document}</tex-math></inline-formula> Under some general assumptions on the nonlinear term <inline-formula><tex-math id="M4">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula>, there exists a constant <inline-formula><tex-math id="M5">\begin{document}$ \alpha_0\ge 0 $\end{document}</tex-math></inline-formula> such that a global minimizer exists for <inline-formula><tex-math id="M6">\begin{document}$ E_\alpha $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M7">\begin{document}$ \alpha&gt;\alpha_0 $\end{document}</tex-math></inline-formula>, and there is no global minimizer with respect to <inline-formula><tex-math id="M8">\begin{document}$ E_{\alpha} $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M9">\begin{document}$ 0&lt;\alpha&lt;\alpha_0. $\end{document}</tex-math></inline-formula> By virtue of concentration-compactness argument and the strict subadditivity of <inline-formula><tex-math id="M10">\begin{document}$ E_\alpha $\end{document}</tex-math></inline-formula>, the strong convergence of minimizing sequence is obtained. Moreover, we present some criteria which determine <inline-formula><tex-math id="M11">\begin{document}$ \alpha_0 = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ \alpha_0&gt;0 $\end{document}</tex-math></inline-formula>, and the existence of global minimizers for <inline-formula><tex-math id="M13">\begin{document}$ E_{\alpha_0}. $\end{document}</tex-math></inline-formula> 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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