Nonresonance below the first eigenvalue for a semilinear elliptic problem
type: Item
instance of: scholarly article
title: Nonresonance below the first eigenvalue for a semilinear elliptic problem
zbMATH Open document ID: 0637.35037
DOI: 10.1007/BF01456841
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164441
review text: Let \(\Omega\) be a bounded open subset in \({\mathbb{R}}^ N \)with smooth boundary \(\partial \Omega\) and let f(u) be a continuous real-valued function on \({\mathbb{R}}\). We consider the semilinear Dirichlet problem (1) \(-\Delta u=f(u)+h(x)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). The nonlinearity is assumed to lie on the left side of the first eigenvalue \(\lambda _ 1\) of -\(\Delta\) in the following sense: (2) \(\limsup _{u\to \pm \infty} 2F(u)/u\) \(2\leq \lambda _ 1\), where \(F(u)=\int ^{u}_{0}f\) is the associated potential. We are interested in the conditions to be imposed on f or F in order that (1) be solvable for any h(x). This was considered by \textit{A. Hammerstein} [Acta Math. 54, 117-176 (1930)] when (2) holds with a strict inequality. We deal with the situation where equality holds in (2). We also consider the case where the nonlinearity depends on x: f(x,u). Our results in this case are related to those of \textit{J. Mawhin}, \textit{J. R. Ward} and \textit{M. Willem} [Arch. Ration. Mech. Anal. 95, 269-277 (1986)].
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4038336
zbMATH Keywords: semilinear Dirichlet problem
zbMATH Keywords: nonresonance
zbMATH Keywords: variational method
zbMATH Keywords: Schwarz rearrangement
zbMATH Keywords: Steiner rearrangement
zbMATH Keywords: density at infinity
zbMATH Keywords: existence
zbMATH Keywords: solvability
author: Djairo G. de Figueiredo
author: J.-P. Gossez
MaRDI profile type: Publication
cites work: Q3876353
cites work: Q3728477
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cites work: Variational methods and semilinear elliptic equations
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cites work: Isoperimetric Inequalities in Mathematical Physics. (AM-27)
OpenAlex ID: W4251618028
description: scientific article; zbMATH DE number 4038336