Concentration of Solutions for a Singularly Perturbed Neumann Problem in non smooth domains

TL;DR

This paper studies solutions to a nonlinear PDE with Neumann boundary conditions in non-smooth domains, proving that solutions concentrate at specific points on the edges of the domain as a parameter tends to zero.

Contribution

It establishes the concentration behavior of solutions for a singularly perturbed Neumann problem in domains with edges, extending previous results to non-smooth geometries.

Findings

Solutions concentrate at points on the edges of the domain boundary.

Concentration occurs as the perturbation parameter approaches zero.

The analysis applies to domains with non-smooth features like edges.

Abstract

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of $\partial\Omega$ on the edges.