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 ${\Omega }_{\epsilon }$ and solutions $u_{\epsilon }$ of \[\begin{cases} -Δu_\varepsilon=1&\text{ in }\ Ω_\varepsilon\\ u_\varepsilon=0&\text{ on }\ \partialΩ_\varepsilon \end{cases}\] such that, for any integer $k\ge 2$, $u_{\epsilon }$ admits at least $k$ maxima points for small enough $\epsilon$. The domain ${\Omega }_{\epsilon }$ is "not far" to be convex in the sense that it is starshaped, the curvature of $\partial {\Omega }_{\epsilon }$ vanishes at exactly $two$ points and the minimum of the curvature of $\partial {\Omega }_{\epsilon }$ goes to $0$ as $\epsilon \to 0$.

Updated version according to the referee's suggestions. To appear in Amer. Jour. of Math

References in corpus (1)

• A note on the critical points of the localization landscape

Cited by in corpus (1)

• The spectral gap to torsion problem for some non-convex domains