Infinitely many new curves of the Fučík spectrum
type: Item
instance of: scholarly article
title: Infinitely many new curves of the Fučík spectrum
zbMATH Open document ID: 1331.35254
author: Riccardo Molle
author: Donato Passaseo
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2015-12-11T00:00:00Z
review text: This paper is devoted to some results on the Fučík spectrum for the Laplace operator, giving new information on its structure. These results show that if \(\Omega \) is a bounded connected domain of \(\mathbb R^N\), then the Fučík spectrum has infinitely many curves asymptotic to the lines \(\{\lambda_1\} \times\mathbb R\) and \(\mathbb R \times \{\lambda_1\}\), where \(\lambda_1\) denotes the first eigenvalue of the operator \(-\Delta\) in \(H^1_0(\Omega )\). If we set \(u^+=\max\{u, 0\}\) and \(u^-=\max\{-u, 0\}\), then the Fučík spectrum is defined as the set \(\Sigma\) of all pairs \((\alpha , \beta ) \in\mathbb R^2\) such that the Dirichlet problem \[ \begin{alignedat}{2} & \Delta u - \alpha u^- + \beta u^+ = 0 &\quad\text{in }& \Omega,\\ & u= 0, &\quad\text{on }&\partial\Omega,\end{alignedat} \] has nontrivial solutions. The main result of this paper may be stated as follows. \(\quad\) { Theorem 1.1} Let \(\Omega \) be a bounded connected domain of \(\mathbb R^N\) with \(N\geq 3\). Then, there exists a nondecreasing sequence \((b_k)_k\) of positive numbers with the following properties. For every positive integer \(k\) and for all \(\beta >b_k\), there exists \(\alpha_{k,\beta} >\lambda_1\) such that the pair \((\alpha_{k,\beta} , \beta )\) belongs to the Fučík spectrum \(\Sigma\). Moreover, for every positive integer \(k, \alpha_{k,\beta}\) depends continuously on \(\beta\) in \(]b_k, +\infty[, \alpha_{k,\beta} <\alpha_{k+1,\beta}\) for all \(\beta >b_{k+1}\) and \(\lim_{\beta\to\infty} \alpha_{k,\beta} =\lambda_1\). \(\quad\) To prove this theorem the authors use a variational method. For all \(\beta >0\), they consider the functional \[ f_\beta (u) =\int_\Omega [|Du|^2-\beta (u^+)^2]\,dx \] constrained on the set \(S=\{u \in H^1_0(\Omega ): \int_\Omega (u^-)^2\,dx=1\}\). For \(\beta >0\) large enough, the eigenfunction \(u\) is obtained as a constrained critical point for \(f_\beta\) on \(S\), while \(\alpha\) arises as the Lagrange multiplier with respect …
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J66
zbMATH DE Number: 6520567
zbMATH Keywords: elliptic operators
zbMATH Keywords: Fučík spectrum
zbMATH Keywords: variational methods
zbMATH Keywords: multiplicity results
zbMATH Keywords: asymptotic behaviours
MaRDI profile type: Publication
full work available at URL: j.anihpc.2014.05.007
OpenAlex ID: W2039865302
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DOI: 10.1016/J.ANIHPC.2014.05.007
description: scientific article; zbMATH DE number 6520567