Virial functional and dynamics for nonlinear Schrödinger equations of local interactions

Our aim is to verify that the functional in the virial identity classifies the dynamics for nonlinear Schrödinger equations of local interactions. In particular, we give a condition under that there exist stable ground states. Our proof of this stability result is based on the ideas in Colin (Ann Inst H Pincaré 23:753–764, 2006) and Shatah (Math Phys 91:313–327, 1983). However, we emphasize that our argument does not use the strict convexity of the H˙1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^{1}$$\end{document}-norm of ground state with respect to ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}: a key lemma is Lemma 4.8 below. Furthermore, we discuss the limiting profile of ground states (see Theorem 4.4).