Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$
Yibin Zhang
TL;DR
The paper studies boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$, constructing positive solutions with an arbitrary number of mixed interior and boundary bubbles as $\lambda\to0$. Using a Lyapunov–Schmidt reduction, it builds a refined approximate bubble configuration and carries out a detailed linear and nonlinear analysis to reduce the problem to a finite-dimensional energy $F_\lambda$, whose critical points yield true solutions. A precise expansion of the energy in terms of the anisotropic Green function $G_a$ and its regular part $H_a$ allows a degree-theoretic argument to locate solutions concentrating at strict local extrema of $a$ on $\partial\Omega$ or clustering at a single boundary point, extending known isotropic results and connecting to chemotaxis models. The results provide a rigorous framework for understanding how anisotropy and boundary geometry drive boundary-layer and interior-layer bubbling in two dimensions, with explicit asymptotics governing bubble interactions. Overall, the work significantly advances the theory of boundary concentration in anisotropic elliptic problems and offers tools applicable to related higher-dimensional and symmetric settings.
Abstract
Given a smooth bounded domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=λa(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \,\qquad\quad\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $λ>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overlineΩ$ and $ν$ denotes the outer unit normal vector to $\partialΩ$. 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 $\partialΩ$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overlineΩ$ as $λ\rightarrow0$.