Multiple solutions of semilinear Dirichlet problems on a domain with \(S^1\)-group actions
type: Item
instance of: scholarly article
title: Multiple solutions of semilinear Dirichlet problems on a domain with \(S^1\)-group actions
zbMATH Open document ID: 0892.35057
author: J. Ning
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-14T00:00:00Z
review text: The author considers the Dirichlet problem for the non-homogeneous semilinear elliptic equation \[ \Delta u + g(u) = h, \tag{P} \] on \(S^1\)-symmetric domains in \(\mathbb R^N\). The author supposes that the nonlinearity always crosses at least 3 eigenvalues of the corresponding linear boundary value problem. When \(\| h\| _{L^2}\) is small and \(S^1\)-invariant, the existence of at least \(3\) solutions is proved. When \(\| h\| _{L^2}\) is small and not \(S^1\)-invariant, the existence of at least \(5\) solutions is proved. In the proof, the author uses Morse theory and topological degree methods. This result is motivated by results of \textit{A. Ambrosetti} and \textit{G. Mancini} [Nonlinear Anal., Theory Methods Appl. 3, 635-645 (1979; Zbl 0433.35025)], and \textit{L. Boccardo} and \textit{T. Gallouët} [Nonlinear Anal., Theory Methods Appl. 14, No. 1, 81-91 (1990; Zbl 0713.35029)], who proved the existence of exactly \(3\) solutions when the nonlinearity crosses precisely one corresponding linear eigenvalue. This problem is related to a general conjecture on the relations of the number of linear eigenvalues crossed by the nonlinearity and the number of solutions of the corresponding nonlinear problem.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1123524
zbMATH Keywords: symmetric domain
zbMATH Keywords: crossing of eigenvalues
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5778
OpenAlex ID: W2063829322
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DOI: 10.1006/JMAA.1997.5778
description: scientific article; zbMATH DE number 1123524