Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions
Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions
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DOI：
10.1016/j.jde.2012.08.001
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发表时间：
2013-01
影响因子：
2.4
通讯作者：
Yasuhito Miyamoto;K. Yagasaki
中科院分区：
文献类型：
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作者：
Yasuhito Miyamoto;K. Yagasaki
设 B⊂Rn(n⩾1) 为单位球，D⊂Rm 为有界域。我们研究了由椭圆诺依曼问题的内部单峰解组成的全局分支以及大 λ 沿分支的第一特征值的单调性，其中当将域替换为 B×D 时，利用这种单调性，我们表明集中于 {0}×D 的解的分支具有二次分岔点。对于 n=1 且 p⩾2 为整数，我们通过显示时间图（周期函数）的单调性来确定全局分岔图。对于 n=1 和 p=3，也证明了沿整个分支的第一特征值的单调性。
Let B⊂Rn(n⩾1) be a unit ball and D⊂Rmbe a bounded domain. We study the global branch consisting of interior single-peak solutions of the elliptic Neumann problem and the monotonicity of the first eigenvalue along the branch for large λ, where When the domain is replaced with B×D, using this monotonicity, we show that the branch of solutions concentrating on {0}×D has secondary bifurcation points. For n=1 and p⩾2 an integer, we determine the global bifurcation diagram by showing the monotonicity of the time-map (the period function). The monotonicity of the first eigenvalue along the whole branch is also proved for n=1 and p=3.