Well-posedness in weighted spaces for the generalized Hartree equation with p < 2

We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

A weak KAM approach to the periodic stationary Hartree equation

We present, through weak KAM theory, an investigation of the stationary Hartree equation in the periodic setting. More in details, we study the Mean Field asymptotics of quantum many body operators thanks to various integral identities providing the energy of the ground state and the minimum value of the Hartree functional. Finally, the ground state of the multiple-well case is studied in the semiclassical asymptotics thanks to the Agmon metric.

Energy scattering for the focusing fractional generalized Hartree equation

<p style='text-indent:20px;'>This note studies the asymptotics of radial global solutions to the non-linear fractional Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\dot u-(-\Delta)^s u+|u|^{p-2}(I_\alpha *|u|^p)u = 0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Indeed, using a new method due to Dodson-Murphy [<xref ref-type="bibr" rid="b10">10</xref>], one proves that, in the inter-critical regime, under the ground state threshold, the radial global solutions scatter in the energy space.</p>

On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

<p style='text-indent:20px;'>We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>. Firstly we study the existence and stability of the maximal ground state <inline-formula><tex-math id="M3">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M4">\begin{document}$ N = N_c $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ N_c $\end{document}</tex-math></inline-formula> is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states <inline-formula><tex-math id="M6">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \beta\rightarrow 0^+ $\end{document}</tex-math></inline-formula>, and the optimal blow-up rate with respect to <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> will be calculated.</p>