{"title":"Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems","authors":"Romulo D. Carlos, Giovany M. Figueiredo","doi":"10.1007/s00526-024-02734-4","DOIUrl":null,"url":null,"abstract":"<p>We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by </p><span>$$\\begin{aligned} \\Delta ^{2} u \\pm \\Delta _p u = f(u) + \\beta |u|^{2_{**}-2}u\\ \\text{ in } \\ \\Omega \\ \\ \\text{ and } \\ \\Delta u=u=0 \\ \\text{ on } \\ \\ \\partial \\Omega , \\end{aligned}$$</span><p>where <span>\\(\\Omega \\subset \\mathbb {R}^{N}\\)</span> is a bounded and smooth domain, <span>\\(2< p\\le \\frac{2N}{N-2}\\)</span> for <span>\\(N\\ge 3\\)</span>, <span>\\(2_{**}=\\frac{2N}{N-4}\\)</span> if <span>\\(N\\ge 5\\)</span>, <span>\\(2_{**}=\\infty \\)</span> if <span>\\(3\\le N <5\\)</span> and <i>f</i> is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case <span>\\(\\beta =0\\)</span> and the critical case <span>\\(\\beta =1\\)</span>.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00526-024-02734-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by
$$\begin{aligned} \Delta ^{2} u \pm \Delta _p u = f(u) + \beta |u|^{2_{**}-2}u\ \text{ in } \ \Omega \ \ \text{ and } \ \Delta u=u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$
where \(\Omega \subset \mathbb {R}^{N}\) is a bounded and smooth domain, \(2< p\le \frac{2N}{N-2}\) for \(N\ge 3\), \(2_{**}=\frac{2N}{N-4}\) if \(N\ge 5\), \(2_{**}=\infty \) if \(3\le N <5\) and f is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case \(\beta =0\) and the critical case \(\beta =1\).
我们考虑以下一类椭圆基尔霍夫-布西内斯克(Kirchhoff-Boussinesq)类型问题,其给定条件为 $$\begin{aligned}\u = f(u) + \beta |u|^{2_{**}-2}u \text{ in }\\Omega \text{ and }.\ Omega (和)\ Delta u=u=0 on }\ \partial \Omega , \end{aligned}$ 其中 \(\Omega \subset \mathbb {R}^{N}\) 是一个有界的光滑域, \(2<;如果 \(N\ge 3\),\(2_{**}=\frac{2N}{N-4}\) if \(N\ge 5\),\(2_{**}=\infty \) if \(3\le N <5\) and f is a continuous function.我们利用内哈里流形上的最小化技术、山口定理和属理论证明了非小解的存在性和多重性。本文考虑了亚临界情况(\beta =0)和临界情况(\beta =1)。