度量图上Lane--Emden问题最小能量解的渐近行为
Asymptotic Behavior of Least Energy Solutions to the Lane--Emden Problem on Metric Graphs
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中文总结 AI 辅助
本文研究紧度量图上Lane--Emden问题最小能量解在$p\to\infty$时的渐近行为，刻画了两种边界条件下的极限变分问题，并证明解收敛到由格林函数或最大距离点对决定的极限轮廓。
中文摘要 AI 辅助
本文研究了当$p \to \infty$时，紧度量图上Lane--Emden问题最小能量解的渐近行为。对于Dirichlet--Kirchhoff边界条件下的问题，我们利用Dirichlet--Kirchhoff格林函数刻画了极限变分问题，并证明最小能量正解（在子序列意义下）收敛到以对角格林函数最大点为心的归一化格林函数。我们还证明其最大点趋近于这类最大点的集合。对于Kirchhoff--Neumann边界条件下的问题，我们在无环紧度量图上刻画了极限变分问题，并证明最小能量解（在子序列意义下）收敛到与图上实现最大距离的一对点相关联的极限轮廓。此外，其最大点与最小点之间的距离收敛到图上的最大距离。
英文摘要
In this paper, we study the asymptotic behavior of least energy solutions to Lane--Emden problems on compact metric graphs as $p \to \infty$. For the problem with Dirichlet--Kirchhoff boundary conditions, we characterize the limiting variational problem in terms of the Dirichlet--Kirchhoff Green function and show that least energy positive solutions converge, up to a subsequence, to a normalized Green function centered at a maximizer of the diagonal Green function. We also show that their maximum points approach the set of such maximizers. For the problem with Kirchhoff--Neumann boundary conditions, we characterize the limiting variational problem on compact metric graphs without cycles and show that least energy solutions converge, up to a subsequence, to a limiting profile associated with a pair of points realizing the maximal distance on the graph. Moreover, the distance between their maximum and minimum points converges to the maximal distance on the graph.
发表机构
- Osaka Metropolitan University（大阪公立大学）
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