Nonlinear elliptic eigenvalue problems with discontinuities
type: Item
instance of: scholarly article
title: Nonlinear elliptic eigenvalue problems with discontinuities
zbMATH Open document ID: 0934.35112
author: Shouchuan Hu
author: Nikolaos S. Papageorgiou
author: Nikolaos C. Kourogenis
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-04-13T00:00:00Z
review text: Existence of nontrivial solutions to two nonlinear eigenvalue problems with discontinuous nonlinearities is proved in this paper. The first one is given by \(-\Delta_p\in \lambda[f_0(x,u), f_1(x,u)]\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\geq 2\), \(\lambda\) is a real parameter and \(f_0(x,u)\) and \(f_1(x,u)\) are the inferior and superior limits in \(u\) for fixed \(x\). Existence of at least two nontrivial solutions is proved if \(f\) has growth less than \(|u|^a\) with \(0<a<p-1\), is ``sublinear'' at the origin and satisfies some further conditions. The result is proved by using non-smooth critical point theory by Chang and the corresponding version of the Mountain Pass Theorem. The second one is an associated semilinear elliptic problem. Here the existence, under different assumptions, of at least one solution follows from a linking theorem. We point out that, contrary to the claim of the authors that ``in all these works existence of only one nontrivial solution (eigenvalue) is proved'', page 407, existence of at least two nontrivial positive solutious is proved for \(p=2\) in Theorems 4.1 and 4.2 of [\textit{K. C. Chang}, Sci. Sin. 21, 139-158 (1978; Zbl 0397.35057)]. However, the method of proof (namely the fixed point index) is different.
reviewed by: Jesús Hernández
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1289970
zbMATH Keywords: discontinuous nonlinearity
zbMATH Keywords: multiple solutions
zbMATH Keywords: non-smooth critical point theory
zbMATH Keywords: \(p\)-Laplacian
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6338
OpenAlex ID: W2071727043
cites work: Dual variational methods in critical point theory and applications
cites work: Q4606219
cites work: Variational methods for nonlinear elliptic eigenvalue problems
cites work: Q4180686
cites work: Variational methods for non-differentiable functionals and their applications to partial differential equations
cites work: Q5181909
cites work: Critical point theory for nondifferentiable functionals and applications
cites work: Pseudomonotone operators and nonlinear elliptic boundary value problems
cites work: On the Equation div( | ∇u | p-2 ∇u) + λ | u | p-2 u = 0
cites work: Q4175937
cites work: Elliptic eigenvalue problems with discontinuous nonlinearities
cites work: Q5516356
cites work: Q3749777
cites work: Q4332942
cites work: Q3995594
DOI: 10.1006/JMAA.1999.6338
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description: scientific article; zbMATH DE number 1289970