一类大指数椭圆型Neumann问题的边界层分离和聚集正解
Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent
摘要：
给定$\mathbb{R}^N$中一个光滑有界区域$\mathcal{D}$，其中$p>1$为大指数，$ν$为边界$\partial\mathcal{D}$的外单位法向量，研究了椭圆型Nynann问题$$\begin{cases}-Δ\upsilon+\upsilon=\upsilon^p,\,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\\[1mm] \frac{\partial \upsilon}{\partialν}=0\qquad\qquad\qquad\qquad \textrm{on}\ \partial\mathcal{D}, \end{cases}$$正解的存在性和轮廓.对于适当的整环$\mathcal{D}$，我们用构造性的方法证明了：对于任何非负整数$k$，$k+l\geq1$，如果$p$足够大，则该问题存在一族正解，其边界层和内层都集中在$\partial\mathcal{D}$的$(N-2)$维极小子流形上，或者坍缩到与$p\rightarrow+\infty$相同的$\partial\mathcal{D}$的$(N-2)$维极小子流形。
英文摘要：
Given a smooth bounded domain $\mathcal{D}$ in $\mathbb{R}^N$ with $N\geq3$, we study the existence and the profile of positive solutions for the following elliptic Nenumann problem $$\begin{cases}-Δ\upsilon+\upsilon=\upsilon^p,\,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\\[1mm] \frac{\partial \upsilon}{\partialν}=0\qquad\qquad\qquad\qquad \textrm{on}\ \partial\mathcal{D}, \end{cases}$$ where $p>1$ is a large exponent and $ν$ denotes the outer unit normal vector to the boundary $\partial\mathcal{D}$. For suitable domains $\mathcal{D}$, by a constructive way we prove that, for any non-negative integers $k$, $l$ with $k+l\geq1$, if $p$ is large enough, such a problem has a family of positive solutions with $k$ boundary layers and $l$ interior layers which concentrate along $k+l$ distinct $(N-2)$-dimensional minimal submanifolds of $\partial\mathcal{D}$, or collapse to the same $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $p\rightarrow+\infty$.
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