Nondegeneracy of elliptic problems with periodic nonlinearities in the plane
type: Item
instance of: scholarly article
title: Nondegeneracy of elliptic problems with periodic nonlinearities in the plane
zbMATH Open document ID: 1149.35022
author: Antonio J. Ureña
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2007-06-14T00:00:00Z
review text: Let \(\Omega \subset\mathbb R^N\) be a bounded domain with a smooth boundary and let \(\mathcal{S}_0 = \{ \widetilde{h} \in L^2(\Omega): \int_\Omega \widetilde{h} (\omega)\varphi(\Omega) \,d\omega = 0\}\), with \(\varphi\) being a generator of the one dimensional eigenspace of the first eigenvalue \(\mu_1\) of the Laplace operator with homogeneous Dirichlet boundary conditions. The author investigates qualitative results of existence of solutions for the following semilinear, strongly resonant, elliptic boundary value problem \[ -\Delta u - \mu_1 u + g(u) = h \quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial \Omega, \tag{P} \] where the \(C^1\) nonlinearity \(g\) is assumed to be nontrivial, \(2\pi\)-periodic and to have zero mean value. Denote by \(S_g\) the set of the \(\widetilde{h} \in L^2(\Omega)\) such that the problem (P) is solvable. The A. proves the following result: Let \(N=2\); then the set \(S_g\) contains the hyperplane \(S_0\) into its \(L(\Omega)\)-interior. The result is an extension of a previous result by \textit{S. Solimini} [J. Math. Anal. Appl. 117, 138--152 (1986; Zbl 0634.35030)] to the planar case.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 5164772
zbMATH Keywords: variational methods
zbMATH Keywords: boundary value problems
zbMATH Keywords: nonlinear elliptic problems
reviewed by: Marco Biroli
MaRDI profile type: Publication
full work available at URL: s00033-006-0077-2
OpenAlex ID: W2169706732
DOI: 10.1007/S00033-006-0077-2
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description: scientific article; zbMATH DE number 5164772