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Rearrangement inequalities and duality theory for a semilinear elliptic variational problem. (English) Zbl 0614.49014
L’auteur considère le problème \(-\Delta u-\lambda u=r(x)f(u)\) sur un domaine \(\Omega \subset {\mathbb{R}}^ N\) qui est Steiner symétrique par rapport à des sous-espaces \(J_ 1,J_ 2,...,J_ k\), avec \(u=0\) sur \(\partial \Omega\) et \(f(u)\sim u^{p-1}\), \(2<p<2N/(N-3)\), \(\lambda <\lambda_ 0=la\) première valeur propre de -\(\Delta\). Il établit l’existence d’une solution qui est symétrique décroissante par rapport à \(J_ 1,J_ 2,...,J_ k\). La démonstration utilise une méthode de dualité introduite par J. Toland [Arch. Ration. Mech. Anal. 71, 41-61 (1979; Zbl 0411.49012)].
Reviewer: H.Brezis

MSC:
49N15 Duality theory (optimization)
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J20 Variational methods for second-order elliptic equations
42C20 Other transformations of harmonic type
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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