Multi-peak solutions for a class of nonlinear Schrödinger equations

In this paper we consider the study of positive solutions of¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ -\varepsilon^2\Delta u+\lambda u=f(x,u)\quad {\rm on}\quad \mathbb{R}^N, $\end{document}¶¶where ε is a small parameter, λ>0 and f is an appropriate function. Here we find multi-peak solutions exhibiting concentration at any prescribed "stable" set of zeroes of the field¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\cal S}(P)=\int\limits_{\mathbb{R}^N}\left[\nabla_xf(P,U_P(y))\cdot y\right]\nabla U_P(y)dy,\quad P\in \mathbb{R}^N, $\end{document}¶¶where UP is the unique radial solution of the limit equation¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ -\Delta U_P+\lambda U_P=f(P,U_P)\quad {\rm on} \quad \mathbb{R}^N. $\end{document}¶¶Conversely, we show that the points at which a sequence of multi-peak solutions concentrate must be zeroes of the field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\cal S} $\end{document}.