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 $N\ge 3$, we study the existence and the profile of positive solutions for the following elliptic Nenumann problem $$\{\begin{array}{l}-\Delta \upsilon +\upsilon ={\upsilon }^p,\upsilon >0\text{in}\mathcal{D},\\ \frac{\partial \upsilon }{\partial \nu }=0\text{on}\partial \mathcal{D},\end{array}$$ 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 $k+l\ge 1$, 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 $p\to +\infty$.

arXiv commentThis manuscript has been accepted for publication in Communications in Contemporary Mathematics