On the solutions of a singular elliptic equation concentrating on a circle

TL;DR

This paper studies solutions to a singularly perturbed elliptic equation in an annulus that concentrate on a circle as the perturbation parameter approaches zero, revealing how the concentration depends on a parameter.

Contribution

It introduces a reduction method to analyze concentration phenomena of solutions on a circle in a singular elliptic problem, linking the concentration to the parameter lpha.

Findings

Existence of solutions concentrating on a circle as ps .

The concentration profile depends explicitly on the parameter lpha.

Reduction to a lower-dimensional problem simplifies analysis.

Abstract

Let $A=\{x\in \R^{2N+2} : 0< a< |x| <b\}$ be an annulus. Consider the following singularly perturbed elliptic problem on $A$ \begin{equation} \begin{array}{lll} -\eps^2{\De u} + |x|^{\alpha}u = |x|^{\alpha}u^p, &\mbox{\qquad in} A \notag u>0 &\mbox{\qquad in} A \frac{\partial u}{\partial\nu} = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^*-1$. We shall show that there exists a positive solution $u_\eps$ concentrating on an $S^1$ orbit as $\eps\to 0$. We prove this by reducing the problem to a lower dimensional one and analyzing a single point concentrating solution in the lower dimensional space. We make precise how the single peak concentration depends on the parameter $\alpha$.