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Classification of the Solutions of Semilinear Elliptic Problems in a Ball

Benguria, Rafael; Dolbeault, Jean; Esteban, Maria J. (2000), Classification of the Solutions of Semilinear Elliptic Problems in a Ball, Journal of Differential Equations, 167, 2, p. 438-466. http://dx.doi.org/10.1006/jdeq.2000.3792

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Type
Article accepté pour publication ou publié
Date
2000
Nom de la revue
Journal of Differential Equations
Volume
167
Numéro
2
Éditeur
Elsevier
Pages
438-466
Identifiant publication
Métadonnées
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Auteur(s)
Benguria, Rafael
Dolbeault, Jean cc
Esteban, Maria J. cc
Résumé (EN)
In this paper we fully describe the set of the positive and nodal (regular and singular) radial solutions of the superlinear elliptic PDE. −Δu=λu+|u|p−1 u in B1, u=0 on ∂B1, p>1, (1)without restriction on the range of λset membership, variantImage . Here, B1 is the unit ball in Image N. More precisely, in all subcritical, critical and supercritical cases, we analyze the possible singularities of radial solutions at the origin and the number of bounded and unbounded solutions. The solutions will be of three different types: bounded with a finite number of zeroes in (0, 1), singular at the origin, still with a finite number of zeroes and singular with sign changing oscillations at the origin.
Mots-clés
nodal solutions; oscillatory solutions; multiplicity branches; bifurcations; critical exponent; Pohozaev's identity; semilinear elliptic equations; removable singularities