On the number and location of critical points of solutions of nonlinear elliptic equations in domains with a small hole

Abstract

In this paper we study the following problem \begin{equation} \begin{cases} -Δu=f(u)~&\mbox{in}\ Ω_\varepsilon,\\ u>0~&\mbox{in}\ Ω_\varepsilon,\\ u=0~&\mbox{on}\ \partialΩ_\varepsilon, \end{cases} \end{equation} where ${\Omega }_{\epsilon }=\Omega \backslash B(P,\epsilon )$, $\Omega \subset R^N$ with $N\ge 2$ is a smooth bounded domain, $B(P,\epsilon )$ is the ball centered at $P$ and radius $\epsilon >0$ and $f$ is a smooth nonlinearity. By some computations involving the Green function and degree theory, we compute the number and location of critical points of solutions for small $\epsilon >0$.