Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth.
type: Item
instance of: scholarly article
title: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth.
zbMATH Open document ID: 1274.35087
author: Giovany M. Figueiredo
author: João R. Santos Júnior
published in: Differential and Integral Equations
publication date: 2013-04-09T00:00:00Z
review text: In this article, the existence of infinitely many weak solutions for the Dirichlet problem involving nonlocal equations of Kirchhoff type is investigated. In particular, the authors consider the following problem: \[ - M\left (\int _{\Omega }| \nabla u| ^2 \text{d}x \right ) \Delta u = \lambda | u| ^{q-2}u +| u| ^{p-2}u \, \text{ in } \Omega ,\; u=0 \text{ on } \partial \Omega , \] where \(\Omega \) is a smooth domain in \(\mathbb {R}^N\), with \(N\in \{1,2,3\}\), \(M\:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) is a continuous function, \(\lambda \) is a positive parameter, and the exponents \(q,p\) are such that \(1<q<2<p\), with \(p\leq 6\) if \(N=3\). For \(N=3\), the subcritical cases \(2<p<4\), \(p=4\), \(4<p<6\) and the critical case \(p=6\) are singly examinated and related existence results of infinitely many weak solutions are obtained under suitable conditions on the function \(M\). The existence of infinitely many weak solutions is established for all \(\lambda >0\) when \(2<p\leq 4\), and for all \(\lambda >0\) small enough when \(4<p\leq 6\). The proofs follow standard arguments based on variational methods. The key tool is a consequence of a well known abstract result by \textit{D. C. Clark} [Math. J., Indiana Univ. 22, 65--74 (1972; Zbl 0228.58006)] on the existence of infinitely critical points for \(C^1\)-functionals on Banach spaces which satisfy the Palais-Smale condition and are invariant under the action of a compact topological group.
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 6152276
zbMATH Keywords: nonlocal elliptic problem
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: variational method
zbMATH Keywords: weak solution
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6152276