Existence of infinitely many weak solutions for a Neumann problem
type: Item
instance of: scholarly article
title: Existence of infinitely many weak solutions for a Neumann problem
zbMATH Open document ID: 1058.35071
author: Giovanni Anello
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: The paper is concerned with the Neumann problem \[ -\Delta_p u= \mu f(x,u)+h(x,u), \quad \text{in } \Omega, \] \[ \partial u/\partial\nu=0 \quad \text{on } \partial\Omega, \] on a bounded \(C^1\)-domain \(\Omega\subset\mathbb R^N\). Here \(\Delta_p\) is the \(p\)-Laplace operator, \(p>1\), \(\mu\) is a parameter, and \(f,h:\Omega\times\mathbb R\to\mathbb R\) are nonlinear functions. The function \(f(x,t)\) is of order \(O(| t|^{p-1})\) as \(| t|\to\infty\), and \(h(x,t)=O(| t|^q)\) for some \(q<(p-1)N+p(N-p)\). The main theorem states that under an additional condition on \(H(x,t)=\int^t_0 h(x,s)\,ds\), there exists \(\mu^*\in(0,\infty]\) such that \((P_\mu)\) for \(|\mu|<\mu^*\) has an unbounded sequence of solution in \(W^{1,p}(\Omega)\). As an example, the theorem applies with \(\mu^*=\infty\) if \(H(x,t)=-\lambda(x)| t|^m g(\text{{ln}}| t|)\) with \(\lambda\in L^\infty(\mathbb R)\), \(\text{{essinf}}\,\lambda>0\), \(p<m 0\). The proof uses variational methods and is based on a critical point theorem of \textit{B. Ricceri} [J. Comput. Appl. Math. 113, 401--410 (2000; Zbl 0946.49001)].
reviewed by: Thomas J. Bartsch
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2102567
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Neumann problem
zbMATH Keywords: variational methods
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.009
OpenAlex ID: W2060555316
cites work: INFINITELY MANY SOLUTIONS TO THE NEUMANN PROBLEM FOR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN AND WITH DISCONTINUOUS NONLINEARITIES
cites work: Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the \(p\)-Laplacian
cites work: A general variational principle and some of its applications
cites work: INFINITELY MANY SOLUTIONS OF THE NEUMANN PROBLEM FOR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN
cites work: Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems
DOI: 10.1016/J.NA.2004.02.009
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description: scientific article; zbMATH DE number 2102567