Asymptotic Behavior of Least Energy Solutions to the Lane--Emden Problem on Metric Graphs
Abstract
In this paper, we study the asymptotic behavior of least energy solutions to Lane--Emden problems on compact metric graphs as p → ∞. 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.
Related
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BibTeX
@misc{sato2026asymptotic,
title = {{Asymptotic Behavior of Least Energy Solutions to the Lane--Emden Problem on Metric Graphs}},
author = {Kazuki Sato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15486},
eprint = {2609.15486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15486},
}
RIS
TY - UNPB
TI - Asymptotic Behavior of Least Energy Solutions to the Lane--Emden Problem on Metric Graphs
AU - Kazuki Sato
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15486
UR - https://arxiv.org/abs/2609.15486
AB - In this paper, we study the asymptotic behavior of least energy solutions to Lane--Emden problems on compact metric graphs as p → ∞. 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.
ER -