Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

Abstract

<p style='text-indent:20px;'>In this paper, we undertake a comprehensive study for existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u+\omega u = a|u|^{p}u +E_1(|u|^{2})u\; \; \; in\; \mathbb{R}^2\; or \; \mathbb{R}^3,\;\;\;\;\;\;{\rm{(DS)}} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which appears in the description of the evolution of surface water waves. In the case of <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-critical case, i.e., <inline-formula><tex-math id="M8">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ a&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ 0&lt;p&lt;2 $\end{document}</tex-math></inline-formula>, we show that normalized ground states blow up as <inline-formula><tex-math id="M11">\begin{document}$ c \nearrow c^*: = \|R\|^2_{L^2} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ R $\end{document}</tex-math></inline-formula> is the ground state solution to equation (DS) with <inline-formula><tex-math id="M13">\begin{document}$ a = 0 $\end{document}</tex-math></inline-formula>. We then give a detailed description for the asymptotic behavior of normalized ground states as <inline-formula><tex-math id="M14">\begin{document}$ c \nearrow c^* $\end{document}</tex-math></inline-formula>. In the case of <inline-formula><tex-math id="M15">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-supercritical case, i.e., <inline-formula><tex-math id="M16">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>, we prove several existence and stability/instability results. We also give new criteria about global existence and blow-up for the associated evolutional equation.

Locations

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