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

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.