Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem
Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem
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二维椭圆问题的莫尔斯指数和某些解的最大点数
DOI：
10.1007/s00013-009-0021-8
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发表时间：
2009
影响因子：
0.6
通讯作者：
F. Takahashi
中科院分区：
文献类型：
--
作者：
F. Takahashi
在本文中，我们考虑问题$$-\Delta u = u^p \quad {\rm{in}}\; \Omega，\quad u > 0 \quad {\rm{in}} \; \Omega，\quad u|_{\partial \Omega} = 0$$on a smooth bounded domain Ω inforp> 1.设上述问题的正解满足莫尔斯指数小于或等于。证明了如果fup进一步满足条件asp → ∞，则对于充分大的fup，其极大值点的个数小于或等于tom。当Ω是凸的时，满足上述假设的莫尔斯指数为1的解存在唯一的临界点，且当p足够大时，水平集是星形的.
In this note, we consider the problem $$-\Delta u = u^p \quad {\rm{in}}\; \Omega, \quad u > 0 \quad {\rm{in}} \; \Omega, \quad u |_{\partial \Omega} = 0$$on a smooth bounded domain Ω inforp> 1. Letupbe a positive solution of the above problem with Morse index less than or equal to. We prove that ifupfurther satisfies the assumptionasp→ ∞, then the number of maximum points ofupis less than or equal tomforpsufficiently large. If Ω is convex, we also show that a solution of Morse index one satisfying the above assumption has a unique critical point and the level sets are star-shaped forpsufficiently large.