Blowup results and concentration in focusing Schrödinger-Hartree equation

Abstract

<p style='text-indent:20px;'>This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in <inline-formula><tex-math id="M1">\begin{document}$ \dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N) $\end{document}</tex-math></inline-formula> and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded <inline-formula><tex-math id="M2">\begin{document}$ \dot{H}^{S_c} $\end{document}</tex-math></inline-formula> norm definitely have concentration properties related to a fixed <inline-formula><tex-math id="M3">\begin{document}$ \dot{H}^{S_c} $\end{document}</tex-math></inline-formula> norm of certain standing waves.