On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Nicola Soave, Giorgio Tortone
{"title":"On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity","authors":"Nicola Soave,&nbsp;Giorgio Tortone","doi":"10.1007/s00205-024-01970-4","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem </p><div><div><span>$$\\begin{aligned} -\\Delta u = \\lambda _+(u^+)^{p-1}-\\lambda _-(u^-)^{q-1}, \\end{aligned}$$</span></div></div><p>where <span>\\(1 \\le p&lt;q&lt;2\\)</span>, <span>\\(\\lambda _+ &gt;0\\)</span>, <span>\\(\\lambda _- \\ge 0\\)</span>. The equation is characterized by the sublinear <i>inhomogeneous</i> character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-01970-4","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

Abstract

We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem

$$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$

where \(1 \le p<q<2\), \(\lambda _+ >0\), \(\lambda _- \ge 0\). The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.

论某些无均质性亚线性方程解的节点集
Abstract We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem $$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$ 其中 \(1 \le p<;q<2\) ,\(\lambda _+ >0\) ,\(\lambda _-\ge 0\) 。该方程的特点是右边具有亚线性非均质性,这使得它很难以标准方式适应自由边界问题的经典工具,如单调性公式和炸毁论证。我们的主要结果是:接近结点集的解的局部行为;可接受消失阶的完整分类,以及对局部最小值奇异集豪斯多夫维度的估计;退化(非局部最小)解的存在。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信