Remarks on a resonant problem with an unbounded nonlinearity
type: Item
instance of: scholarly article
title: Remarks on a resonant problem with an unbounded nonlinearity
zbMATH Open document ID: 0878.35046
author: João Batista de Mendonça Xavier
author: Olimpio H. Miyagaki
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-12-18T00:00:00Z
review text: The semilinear elliptic boundary value problem \[ -\Delta u=K(x)u +g(x,u) \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega, \] is studied, where \(\Omega\) is a smooth, bounded domain in \(\mathbb{R}^N\), \(K\in L^p (\Omega)\), \(p>N/2\) and \(K>0\) in \(\Omega\), with \(g\) a Carathéodory function satisfying \(|g(x,z)|\leq\gamma|s|^\alpha +b(x)\) with \(b\in L^2 (\Omega)\), \(b\geq 0\) and \(0<\alpha\leq l\), \(\gamma>0\). Existence of a solution is proved if \(\mu_\ell (K)=l\) for some \(l=1,2,3 \dots\) (these are the eigenvalues of the linear part) and some integral condition involving the primitive \(G\) of \(g\) is satisfied. Under some additional assumption existence of a nontrivial solution is also proved. These results extend in particular the previous work by \textit{D. Costa} [A note on unbounded perturbations of linear resonant problems, University of Brasilia (1989)]. The first theorem is proved by using the Saddle Point Theorem by Rabinowitz and the second one follows from the Mountain Pass Theorem by Ambrosetti and Rabinowitz.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 1031567
zbMATH Keywords: critical point theory
zbMATH Keywords: existence
zbMATH Keywords: integral condition
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5374
OpenAlex ID: W1969082584
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DOI: 10.1006/JMAA.1997.5374
description: scientific article; zbMATH DE number 1031567