On the solvability of a quasilinear elliptic resonance problem near its first eigenvalue
type: Item
instance of: scholarly article
title: On the solvability of a quasilinear elliptic resonance problem near its first eigenvalue
zbMATH Open document ID: 0931.35051
author: Chung-Cheng Kuo
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-05T00:00:00Z
review text: Let \(\Omega\subset\mathbb{R}^N\), \(N\geq 1\), be a bounded smooth domain and \(\alpha\) an ordered \(N\)-tuple of nonnegative integers. For \(u,v\in W_0^{m,p} (\Omega)\), \(m\geq 1\), \(1<p <\infty\), define the operator \(Q\) by \(Q(u)(v)= \sum_{ |\alpha|\leq m}\int_\Omega A_\alpha D^\alpha v dx\), where the coefficients \(A_\alpha\) are Caratheodory functions which may depend on \(x\) and the derivatives of \(u\) and satisfy appropriate growth conditions. Using Galerkin-type techniques and Brouwer's fixed point theorem, the author studies elliptic resonance problems. Given a Carathéodory function \(f\) and \(G\in(W_0^{m,p}(\Omega))^*\), he proves the existence of a function \(u\in W_0^{m, p}(\Omega)\) such that \[ Q(u)(v) -\mu\int_\Omega (\text{sgn} u)|u|^{p-1}v dx+ \int_\Omega f(x,u) v dx= G(v) \] for all \(v\in W_0^{m,p} (\Omega)\), \(\mu\) being the first eigenvalue of \(Q\). The purpose of this article is to extend previous results to the case that \(f\) has no growth restriction as \(|u|\to\infty\) and in which the Landesman-Lazer condition is not satisfied.
reviewed by: Ralf Beyerstedt
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1309048
zbMATH Keywords: quasilinear elliptic resonance problem
zbMATH Keywords: Landesman-Lazer condition
zbMATH Keywords: Galerkin-type techniques
zbMATH Keywords: Brouwer's fixed point theorem
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6328
OpenAlex ID: W1994572700
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cites work: Quasilinear ellipticity and the ist eigenvalue
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DOI: 10.1006/JMAA.1999.6328
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description: scientific article; zbMATH DE number 1309048