Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Romulo D. Carlos, Giovany M. Figueiredo
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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 \(\beta =1\).

一类椭圆 Kirchhoff-Boussinesq 类型问题的非微观解的存在性和多重性
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)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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