{"title":"The positive solutions for a class of Kirchhoff-type problems with critical Sobolev exponents on a bounded domain","authors":"Xiaoxue Zhu, Haining Fan","doi":"10.1002/mma.10535","DOIUrl":null,"url":null,"abstract":"<p>We study the positive solutions for a class of Kirchhoff-type problems involving the nonlinearity \n<span></span><math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n <mi>f</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n <msup>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mi>g</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n <msup>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mn>5</mn>\n </mrow>\n </msup>\n <mo>(</mo>\n <mn>2</mn>\n <mo><</mo>\n <mi>p</mi>\n <mo><</mo>\n <mn>4</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$$ \\lambda f(x){u}&amp;#x00026;#x0005E;{p-1}&amp;#x00026;#x0002B;g(x){u}&amp;#x00026;#x0005E;5\\left(2&amp;#x0003C;p&amp;#x0003C;4\\right) $$</annotation>\n </semantics></math> on a bounded domain. The major difficulty of such problems is the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition; especially, we cannot use Pohozaev's identity directly since our domain is bounded and the weight potentials are not \n<span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$$ {C}&amp;#x00026;#x0005E;1 $$</annotation>\n </semantics></math>-smoothness. Another difficulty is caused by the absence of compactness as the appearance of the critical Sobolev growth. In this work, we shall combine the Nehari manifold and some novel analytical skills to overcome the above difficulties and then obtain some existence results. Furthermore, we show some asymptotic behaviors of the solutions.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 4","pages":"4090-4116"},"PeriodicalIF":2.1000,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10535","RegionNum":3,"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 study the positive solutions for a class of Kirchhoff-type problems involving the nonlinearity
on a bounded domain. The major difficulty of such problems is the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition; especially, we cannot use Pohozaev's identity directly since our domain is bounded and the weight potentials are not
-smoothness. Another difficulty is caused by the absence of compactness as the appearance of the critical Sobolev growth. In this work, we shall combine the Nehari manifold and some novel analytical skills to overcome the above difficulties and then obtain some existence results. Furthermore, we show some asymptotic behaviors of the solutions.
期刊介绍:
Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.
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