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Uniqueness of the critical point for semi-stable solutions in $$\mathbb {R}^2$$ R 2
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-01-18 , DOI: 10.1007/s00526-020-01903-5 Fabio De Regibus , Massimo Grossi , Debangana Mukherjee
中文翻译:
$$ \ mathbb {R} ^ 2 $$ R 2中半稳定解的临界点的唯一性
更新日期:2021-01-18
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-01-18 , DOI: 10.1007/s00526-020-01903-5 Fabio De Regibus , Massimo Grossi , Debangana Mukherjee
In this paper we show the uniqueness of the critical point for semi-stable solutions of the problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=f(u)&{}\quad \text {in }\Omega \\ u>0&{}\quad \text {in }\Omega \\ u=0&{}\quad \text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \subset \mathbb {R}^2\) is a smooth bounded domain whose boundary has nonnegative curvature and \(f(0)\ge 0\). It extends a result by Cabré-Chanillo to the case where the curvature of \(\partial \Omega \) vanishes.
中文翻译:
$$ \ mathbb {R} ^ 2 $$ R 2中半稳定解的临界点的唯一性
在本文中,我们显示了问题的半稳定解的临界点的唯一性
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\ Delta u = f(u)&{} \ quad \ text {in} \ Omega \\ u> 0&{} \ quad \ text {in} \ Omega \\ u = 0&{} \ quad \ text {on} \ partial \ Omega,\ end {array} \ right。} \ end {aligned} $$其中\(\ Omega \ subset \ mathbb {R} ^ 2 \)是一个光滑的有界域,其边界具有非负曲率和\(f(0)\ ge 0 \)。它将Cabré-Chanillo的结果扩展到\(\ partial \ Omega \)的曲率消失的情况。