Positive solutions of an elliptic equation with negative exponent: Stability and critical power☆
Positive solutions of an elliptic equation with negative exponent: Stability and critical power☆
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DOI：
10.1016/j.jde.2008.08.008
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发表时间：
2009-03
影响因子：
2.4
通讯作者：
Yihong Du;Zongming Guo
中科院分区：
文献类型：
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作者：
Yihong Du;Zongming Guo
研究了方程的正解，其中p>0，α>−2，Ω是有界或无界区域.我们证明了存在一个临界幂p=pc（α），使得当p>pc（α）时，Ω= RN的方程没有稳定的正解，但当0<p <pc（α）时，方程存在一族稳定的正解.当p>pc（α−）（α−=min{α，0}）时，我们进一步证明了当Ω = Br BR时，该方程也不存在有限莫尔斯指标的正解.在其他结果中，我们还分类了Br <${0}上在0附近没有界的正解。
We study positive solutions of the equation where p>0, α>−2, and Ω is a bounded or unbounded domain. We show that there is a critical power p=pc(α) such that this equation with Ω=RNhas no stable positive solution for p>pc(α) but it admits a family of stable positive solutions when 0<p⩽pc(α). If p>pc(α−)(α−=min{α,0}), we further show that this equation with Ω=Br∖{0} has no positive solution with finite Morse index that has an isolated rupture at 0, and analogously it has no positive solution with finite Morse index when Ω=RN∖BR. Among other results, we also classify the positive solutions over Br∖{0} which are not bounded near 0.