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

2019/04/05 by Yibin Zhang, Zhang, Yibin
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1904.02936

This manuscript has been accepted for publication in Communications in Contemporary Mathematics

openalex publication_date 2019/04/05 · arxiv created 2021/10/09 · arxiv updated 2021/10/12 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28

Abstract

Given a smooth bounded domain D in ℝ^N with N≥3, we study the existence and the profile of positive solutions for the following elliptic Nenumann problem \begincases-Δυ+υ=υ^p, υ>0 \textrmin D,
(∂ υ)/(∂ν)=0 \textrmon \partialD, \endcases where p>1 is a large exponent and ν denotes the outer unit normal vector to the boundary \partialD. For suitable domains D, by a constructive way we prove that, for any non-negative integers k, l with k+l≥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 \partialD, or collapse to the same (N-2)-dimensional minimal submanifold of \partialD as p→+∞.

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