Concentration of solutions for a singularly perturbed mixed problem in non smooth domains

TL;DR

This paper studies the existence and concentration behavior of solutions to a singularly perturbed boundary value problem with mixed boundary conditions in non-smooth domains, revealing how solutions localize near boundary singularities.

Contribution

It establishes the existence of solutions concentrating near boundary intersections in non-smooth domains under geometric conditions, extending prior results to more complex boundary geometries.

Findings

Solutions concentrate near boundary intersections as perturbation parameter tends to zero

Existence of solutions depends on geometric conditions of the domain boundary

Results apply to nonlinear problems with subcritical exponents

Abstract

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $\Omega\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.