{"title":"Periodic solution for Hamiltonian type systems with critical growth","authors":"Yuxia Guo, Shengyu Wu, Shusen Yan","doi":"10.1007/s00526-024-02770-0","DOIUrl":null,"url":null,"abstract":"<p>We consider an elliptic system of Hamiltonian type in a strip in <span>\\({\\mathbb {R}}^N\\)</span>, satisfying the periodic boundary condition for the first <i>k</i> variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on <i>k</i>. The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green’s function of the Laplacian operator in a strip with periodic boundary conditions is found.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"16 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Calculus of Variations and Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00526-024-02770-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider an elliptic system of Hamiltonian type in a strip in \({\mathbb {R}}^N\), satisfying the periodic boundary condition for the first k variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on k. The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green’s function of the Laplacian operator in a strip with periodic boundary conditions is found.
我们考虑了在\({\mathbb {R}}^N\) 带中的哈密顿型椭圆系统，该系统满足前 k 个变量的周期性边界条件。在临界增长的超线性情况下，我们证明了在 k 的最优条件下该系统存在一个单一的冒泡解。本文的新颖之处在于，一旦找到了具有周期性边界条件的条带中拉普拉斯算子的格林函数，就可以得到证明存在结果所需的所有估计值。
期刊介绍：
Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives.
This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include:
- Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory
- Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems
- Variational problems in differential and complex geometry
- Variational methods in global analysis and topology
- Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems
- Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions
- Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.
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