Concentration results for solutions of a singularly perturbed elliptic system with variable coefficients

TL;DR

This paper investigates how solutions to a singularly perturbed elliptic system with variable coefficients concentrate around points in a domain, analyzing the influence of coefficients on the concentration behavior and solutions on higher-dimensional orbits.

Contribution

It establishes the existence of point concentrating solutions for the system and explores how coefficients affect the concentration profile, including solutions on higher-dimensional orbits.

Findings

Existence of point concentrating solutions established.

Coefficients influence the concentration profile.

Applications to solutions on higher-dimensional orbits.

Abstract

In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in } \Omega \newline u>0, \ v>0 \text{ in } \Omega, &\text{ and }\quad\frac{\partial u}{\partial\nu} = 0 = \frac{\partial v}{\partial\nu} &&\text{on }\partial\Omega \end{aligned} \right. \end{equation} where $\Omega$ is a smooth bounded domain in $R^n, n\geq 3$. The coefficients $a(x), b(x)$ and $c(x)$ are positive bounded smooth functions. We shall study the existence of point concentrating solutions and discuss the role of the coefficients to determine the concentration profile of the solutions. We have also discussed some applications of our main theorem towards the existence of solutions concentrating on higher-dimensional orbits.