no code implementations • 18 Feb 2021 • Massimo Grossi, Isabella Ianni, Peng Luo, Shusen Yan
We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -\Delta u=u^{p} &{\text{in}~\Omega},\\[0. 5mm] u>0 &{\text{in}~\Omega},\\[0. 5mm] u=0 &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $\Omega\subset \mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large.
Analysis of PDEs
no code implementations • 29 Jan 2021 • Fabio De Regibus, Massimo Grossi
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 $k\ge2$.
Analysis of PDEs
