Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$

Stuart, C.A.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1259-1281 / Harvested from Numdam

Résumé

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation $ G(u)=\lambda u$, where G acts between real Hilbert spaces and $ \lambda \in \mathbb{R}$, and then for square-integrable solutions of a second order non-linear elliptic equation on $ {\mathbb{R}}^{N}$. The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the H-asymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the H-asymptotic derivative if it is too close to the essential spectrum.

Détail

Publié le : 2015-01-01

MR 3425262 | Zbl 1330.35187

DOI : https://doi.org/10.1016/j.anihpc.2014.09.003
Classification:  35J91,  47J15

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