Nonresonance conditions on the potential with respect to the Fučík spectrum for semilinear Dirichlet problems
type: Item
instance of: scholarly article
title: Nonresonance conditions on the potential with respect to the Fučík spectrum for semilinear Dirichlet problems
zbMATH Open document ID: 1232.35071
author: Xiaojun Chang
author: Yong Li
author: Shuguan Ji
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2011-09-13T00:00:00Z
review text: This paper deals with the following semilinear Dirichlet problem \[ -\Delta u = f(x,u) \,\, \text{ in } \,\, \Omega,\quad u=0\,\, \text{ on } \,\,\partial\Omega, \tag{1} \] where \(\Omega\subset R^m\) (\(m\geq 1\)) is a bounded domain with smooth boundary \(\partial\Omega\), \(f\in C(\overline{\Omega}\times R,R)\). Let \(F(x,u)=\int^u_0 f(x,s)\text{d}s\) be the primitive of \(f(x,u)\), and the Fučík spectrum be defined on the set \(\Sigma\subset R^2\) of points \((\alpha, \beta)\) for which there exists a nontrivial solution of the problem \[ -\Delta u =\alpha u^+-\beta u^- \,\, \text{in} \,\, \Omega,\quad u=0\,\, \text{on} \,\,\partial\Omega. \] The authors establish the existence of solutions of problem (1) by requiring that \(f\) is of linear growth at infinity and the ratio \(\frac{2F(x,s)}{s^2} \) stays asymptotically at infinity away from the Fučík spectrum.
reviewed by: Ramzet M. Dzhafarov
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 5947369
zbMATH Keywords: Fučík spectrum
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: jumping nonlinearity
zbMATH Keywords: nonresonance condition
MaRDI profile type: Publication
full work available at URL: s00033-009-0051-x
OpenAlex ID: W2040029992
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DOI: 10.1007/S00033-009-0051-X
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description: scientific article; zbMATH DE number 5947369