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arXiv · 2101.12652

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

Fabio De Regibus·Massimo Grossi
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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 $\varepsilon\to0$.

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Fabio De Regibus, Massimo Grossi. 2021-01-29. On the number of critical points of stable solutions in bounded strip-like domains. https://arxiv.org/abs/2101.12652

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