On the number of critical points of solutions of semilinear equations in $\mathbb{R}^2$

Abstract

In this paper we construct families of bounded domains $\Omega_\varepsilon$ and solutions $u_\varepsilon$ of \[\begin{cases} -\Delta u_\varepsilon=1&\text{ in }\ \Omega_\varepsilon\\ u_\varepsilon=0&\text{ on }\ \partial\Omega_\varepsilon \end{cases}\] such that, for any integer $k\ge2$, $u_\varepsilon$ admits at least $k$ maxima points for small enough $\varepsilon$. The domain $\Omega_\varepsilon$ is "not far" to be convex in the sense that it is starshaped, the curvature of $\partial\Omega_\varepsilon$ vanishes at exactly $two$ points and the minimum of the curvature of $\partial\Omega_\varepsilon$ goes to $0$ as $\varepsilon\to0$.