Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$

TL;DR

This paper investigates boundary concentration phenomena for solutions to an anisotropic Neumann problem in two dimensions, constructing solutions with multiple bubbles concentrating at specific boundary points as a parameter tends to zero.

Contribution

It introduces a method to construct solutions with multiple interior and boundary bubbles for an anisotropic Neumann problem, highlighting boundary concentration phenomena.

Findings

Solutions with arbitrarily many bubbles are constructed.

Bubbles concentrate at strict local maxima or minima of the boundary function.

Solutions can accumulate at the same boundary point as the parameter approaches zero.

Abstract

Given a smooth bounded domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=\lambda a(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, \Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\, \qquad\quad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \,\qquad\quad\, \textrm{on}\,\,\, \partial\Omega, \end{cases} $$ where $\lambda>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overline{\Omega}$ and $\nu$ denotes the outer unit normal vector to $\partial\Omega$. Under suitable assumptions on anisotropic coefficient $a(x)$, we construct solutions of this problem with arbitrarily many mixed interior and boundary bubbles which concentrate at totally different strict local maximum or minimal boundary points of $a(x)$ restricted to $\partial\Omega$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overline{\Omega}$ as $\lambda\rightarrow0$.