On the number of critical points of stable solutions in bounded strip-like domains

TL;DR

This paper constructs families of star-shaped and positively curved domains close to strips or cylinders where stable solutions to certain PDEs have multiple critical points, challenging previous assumptions about solution simplicity.

Contribution

It introduces new domain families where stable solutions exhibit multiple critical points, expanding understanding of solution behavior in bounded domains.

Findings

Existence of domains with multiple critical points for stable solutions.

Domains can be star-shaped or have positive mean curvature.

Solutions are close to strips or cylinders as parameters tend to zero.

Abstract

In this paper we show that there exists a family of domains $\Omega_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, such that the $stable$ solution of the problem \[ \begin{cases} -\Delta u= g(u)&\hbox{in }\Omega_\varepsilon\\ u>0&\hbox{in }\Omega_\varepsilon\\ u=0&\hbox{on }\partial\Omega_\varepsilon \end{cases} \] admits $k$ critical points with $k\ge2$. Moreover the sets $\Omega_\varepsilon's$ are star-shaped and "close" to a strip as $\varepsilon\to0$. Next, if $g(u)\equiv1$ and $N\ge3$ we exhibit a family of domain $\Omega_\varepsilon's$ with $positive$ $mean$ $curvature$ and solutions $u_\varepsilon $ which have $k$ critical points with $k\ge2$. In this case, the domains $\Omega_\varepsilon $ turn out to be "close" to a cylinder as $\varepsilon\to0$.