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Nonlinear Eigenvalue Problems for Even Functionals
Stockholm University, Faculty of Science, Department of Mathematics. Stockholm University, Faculty of Science, Department of Mathematics.
2007 (English)In: Applicable Analysis: an international journal, ISSN 1563-504X (electronic) 0003-6811 (paper), Vol. 86, no 7, p. 829-849Article in journal (Refereed) Published
Abstract [en]

Let $H$ be a Hilbert space and let $g\in C^1(H,\mathbb R)$ be an even Fréchet differentiable functional with completely continuous derivative. We introduce maximin values

$\sigma_k(t)$ which are critical values of $g$ restricted to the sphere

$$ S_t = \left\{ u\in H;\; \frac{1}{2} \|u\|^2 = t \right\}$$

and show that the functions $\sigma_k(t)$ have right and left derivatives and that $\sigma_{k\pm}'(t)$ are eigenvalues of g', i.e. there exist $u_k^{\pm}\in S_t$ such that

$$ g'(u_k^\pm) = \sigma_{k\pm}'(t) u_k^{\pm}.

Applications of the result are given to semilinear elliptic equations.

Place, publisher, year, edition, pages
2007. Vol. 86, no 7, p. 829-849
Keyword [en]
nonlinear eigenvalue problem, Krasnoselskii genus, elliptic semilinear equations.
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:su:diva-22159DOI: doi:10.1080/00036810701460503ISI: 000250334600004OAI: oai:DiVA.org:su-22159DiVA, id: diva2:188686
Available from: 2007-12-19 Created: 2007-12-19 Last updated: 2011-01-11Bibliographically approved

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