具有超临界凹凸非线性的kirchhoff型方程的正解

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2025-06-25 DOI:10.1002/mana.70002

Liying Shan, Wei Shuai

{"title":"具有超临界凹凸非线性的kirchhoff型方程的正解","authors":"Liying Shan, Wei Shuai","doi":"10.1002/mana.70002","DOIUrl":null,"url":null,"abstract":"We study the following Kirchhoff-type equation\n\n Via a new variational principle established by Moameni (C. R. Math. Acad. Sci. Paris. 355 (2017) 1236–1241), we shall show that, for each <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>></mo>\n <mn>2</mn>\n </mrow>\n <annotation>$p>2$</annotation>\n </semantics></math>, there exists <math>\n <semantics>\n <mrow>\n <msup>\n <mi>λ</mi>\n <mo>∗</mo>\n </msup>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\lambda ^*>0$</annotation>\n </semantics></math> such that for each <math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <msup>\n <mi>λ</mi>\n <mo>∗</mo>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$\\lambda \\in (0,\\lambda ^*)$</annotation>\n </semantics></math> Equation (0.1) has a positive solution with negative energy. Furthermore, by using the improved Clark theorem, we can obtain a sequence of solutions with negative energy converging to zero in <math>\n <semantics>\n <mrow>\n <msup>\n <mi>L</mi>\n <mi>∞</mi>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$L^{\\infty }(\\Omega)$</annotation>\n </semantics></math> without the restriction of <math>\n <semantics>\n <mi>λ</mi>\n <annotation>$\\lambda$</annotation>\n </semantics></math>.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 8","pages":"2476-2492"},"PeriodicalIF":0.8000,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Positive solution for the Kirchhoff-type equation with supercritical concave and convex nonlinearities\",\"authors\":\"Liying Shan, Wei Shuai\",\"doi\":\"10.1002/mana.70002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the following Kirchhoff-type equation\\n\\n Via a new variational principle established by Moameni (C. R. Math. Acad. Sci. Paris. 355 (2017) 1236–1241), we shall show that, for each <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>></mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$p>2$</annotation>\\n </semantics></math>, there exists <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>λ</mi>\\n <mo>∗</mo>\\n </msup>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$\\\\lambda ^*>0$</annotation>\\n </semantics></math> such that for each <math>\\n <semantics>\\n <mrow>\\n <mi>λ</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <msup>\\n <mi>λ</mi>\\n <mo>∗</mo>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\lambda \\\\in (0,\\\\lambda ^*)$</annotation>\\n </semantics></math> Equation (0.1) has a positive solution with negative energy. Furthermore, by using the improved Clark theorem, we can obtain a sequence of solutions with negative energy converging to zero in <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>L</mi>\\n <mi>∞</mi>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>Ω</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$L^{\\\\infty }(\\\\Omega)$</annotation>\\n </semantics></math> without the restriction of <math>\\n <semantics>\\n <mi>λ</mi>\\n <annotation>$\\\\lambda$</annotation>\\n </semantics></math>.\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"298 8\",\"pages\":\"2476-2492\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.70002\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.70002","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们利用Moameni （C. R. Math）建立的一个新的变分原理来研究下列kirchhoff型方程。学术科学Paris. 355(2017) 1236-1241)，我们将表明，对于每个p ＆gt；2 $p>2$，存在λ∗＆gt；0 $\lambda ^*>0$使得对于每个λ∈(0,λ∗)$\lambda \in (0,\lambda ^*)$方程（0.1）有一个负能量的正解。进一步，利用改进的Clark定理，我们可以在不受λ $\lambda$约束的情况下，得到在L∞（Ω） $L^{\infty }(\Omega)$上收敛于零的负能量解序列。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Positive solution for the Kirchhoff-type equation with supercritical concave and convex nonlinearities

We study the following Kirchhoff-type equation

Via a new variational principle established by Moameni (C. R. Math. Acad. Sci. Paris. 355 (2017) 1236–1241), we shall show that, for each $p > 2$ , there exists $λ^{*} > 0$ such that for each $λ \in (0, λ^{*})$ Equation (0.1) has a positive solution with negative energy. Furthermore, by using the improved Clark theorem, we can obtain a sequence of solutions with negative energy converging to zero in $L^{\infty} (Ω)$ without the restriction of $λ$ .

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index