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

Abstract

In this paper we show that there exists a family of domains ${\Omega }_{\epsilon }\subseteq {\mathbb{R}}^N$ with $N\ge 2$, such that the $stable$ solution of the problem \[ \begin{cases} -Δu= g(u)&\hbox{in }Ω_\varepsilon\\ u>0&\hbox{in }Ω_\varepsilon\\ u=0&\hbox{on }\partialΩ_\varepsilon \end{cases} \] admits $k$ critical points with $k\ge 2$. Moreover the sets ${\Omega }_{\epsilon }^{\prime }s$ are star-shaped and "close" to a strip as $\epsilon \to 0$. Next, if $g(u)\equiv 1$ and $N\ge 3$ we exhibit a family of domain ${\Omega }_{\epsilon }^{\prime }s$ with $positive$ $mean$ $curvature$ and solutions $u_{\epsilon }$ which have $k$ critical points with $k\ge 2$. In this case, the domains ${\Omega }_{\epsilon }$ turn out to be "close" to a cylinder as $\epsilon \to 0$.

20 pages, 1 figure