Boundary concentration phenomena for the higher-dimensional Keller–Segel system

We study the existence of steady states to the Keller–Segel system with linear chemotactical sensitivity function on a smooth bounded domain in RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N,$$\end{document}N≥3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3,$$\end{document} having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable (N-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N-2)$$\end{document}-dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.