论文标题
针对各向异性平面椭圆形的Neumann问题的集中解决方案，其重量和大指数
Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-Hénon weight and large exponent
论文作者
论文摘要
令$ω$成为$ \ m athbb {r}^2 $具有平稳边界的有界域u）+a（x）u = a（x）| x-q |^{2α} u^p，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\，\ textrm {in} u} {\partialν} = 0 \，\，\，\ qquad \ quad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \，\，\，\，\，\，\，\，\，\ textrm {其中$ν$表示外部单位正常向量为$ \partialΩ$，$ q \ in \overlineΩ$，$α\ in（-1，+\ infty）\ setMinus \ setMinus \ mathbb {n} $，$ p> 1 $是一个很大的指数，$ a（x）$是一个积极的光滑功能。我们研究了各向异性系数$ a（x）$和单数来源$ q $之间的相互作用对集中解决方案的存在。我们表明，如果$ q \inΩ$是$ a（x）$的严格局部最大点，则存在一个积极解决方案的家族，任意许多内部峰值积累到$ q $；虽然如果$ q \ in \partialΩ$是$ a（x）$的严格局部最大点，并且满足$ \ langle \ nabla a（q），\，c）\ rangle = 0 $，那么这个问题就有一个积极的解决方案，与任意混合的内部和边界跨越$ q $，有一个积极的解决方案。特别是，我们发现，无论是$ q \ in \overlineΩ$是$ a（x）$或不隔离的本地最大点，是否可以始终以单数来源$ q $的浓度。
Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2α}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $ν$ denotes the outer unit normal vector to $\partialΩ$, $q\in\overlineΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions. We show that if $q\inΩ$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while if $q\in\partialΩ$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,ν(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overlineΩ$ is an isolated local maximum point of $a(x)$ or not.
