Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent

Abstract: 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}-\Delta \upsilon+\upsilon=\upsilon^p,\,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\[1mm] \frac{\partial \upsilon}{\partial\nu}=0\qquad\qquad\qquad\qquad \textrm{on}\ \partial\mathcal{D}, \end{cases}$$ where $p>1$ is a large exponent and $\nu$ 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 $p\rightarrow+\infty$.