Normalized clustering peak solutions for Schrödinger equations with general nonlinearities

We are concerned with the normalized & ell;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-peak solutions to the nonlinear Schr & ouml;dinger equation -epsilon 2 Delta v+V(x)v=f(v)+lambda v,integral RNv2=alpha epsilon N.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon <^>2\Delta v+V(x)v=f(v)+\lambda v,\\ \int _{\mathbb {R}<^>N}v<^>2 =\alpha \varepsilon <^>N. \end{array}\right. } \end{aligned}$$\end{document}Here lambda is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {R}$$\end{document} will arise as a Lagrange multiplier, V has a local maximum point, and f is a general L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2$$\end{document}-subcritical nonlinearity that satisfies a nonlipschitzian property such that lims -> 0f(s)/s=-infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{s\rightarrow 0} f(s)/s=-\infty $$\end{document}. The peaks of solutions that we construct cluster around a local maximum of V as epsilon -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}. Since there is no information about the uniqueness or nondegeneracy of the limiting system, a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of V. We introduce a new method to obtain this estimate, which differs significantly from the ideas of del Pino and Felmer [22] (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka [7, 8] (J. Eur. Math. Soc. 2013 & Mem. Amer. Math. Soc. 2014), where an additional translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case V equivalent to 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\equiv 0$$\end{document}. The ground state energy is not always negative and the strict subadditivity of the ground state energy is achieved here by strict concavity.