关于有界带状区域中稳定解的临界点个数
On the number of critical points of stable solutions in bounded strip-like domains
摘要：
在本文中，我们证明了存在一族域$Ω_{\varepsilon}\subseteq\mathbb{R}^N$与$N\ge2$，使得问题\[ \begin{cases} -Δu= g(u)&\hbox{in }Ω_\varepsilon\\ u>0&\hbox{in }Ω_\varepsilon\\ u=0&\hbox{on }\partialΩ_\varepsilon \end{cases} \]的$stable$解允许$k$临界点与$k\ge2$。此外，集合$Ω_\varepsilon's$是星形的，并且“接近”作为$\varepsilon\to0$的条带。接下来，如果$g(u)\equiv1$和$N\ge3$，我们展示了具有$positive$ $mean$ $curvature$的域$Ω_\varepsilon's$和具有$k$临界点和$k\ge2$的解$u_\varepsilon $的族。在这种情况下，结构域$Ω_\varepsilon $变成“接近”圆柱体，如$\varepsilon\to0$。
英文摘要：
In this paper we show that there exists a family of domains $Ω_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, 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 $k\ge2$. Moreover the sets $Ω_\varepsilon's$ 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 $Ω_\varepsilon's$ with $positive$ $mean$ $curvature$ and solutions $u_\varepsilon $ which have $k$ critical points with $k\ge2$. In this case, the domains $Ω_\varepsilon $ turn out to be "close" to a cylinder as $\varepsilon\to0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
