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

Stuart, C.A.

doi:10.1016/j.anihpc.2014.09.003

Asymptotic bifurcation and second order elliptic equations on

ℝ^{N}

Stuart, C.A.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1259-1281

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Résumé

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation $G (u) = λ u$ , where G acts between real Hilbert spaces and $λ \in ℝ$ , and then for square-integrable solutions of a second order non-linear elliptic equation on $ℝ^{N}$ . 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.

MR Zbl

DOI : 10.1016/j.anihpc.2014.09.003

Classification : 35J91, 47J15
Keywords: Asymptotic linearity, Asymptotic bifurcation, Nonlinear elliptic equation

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     author = {Stuart, C.A.},
     title = {Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1259--1281},
     publisher = {Elsevier},
     volume = {32},
     number = {6},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.09.003},
     mrnumber = {3425262},
     zbl = {1330.35187},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.09.003/}
}

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Stuart, C.A. Asymptotic bifurcation and second order elliptic equations on $ {\mathbb{R}}^{N}$. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1259-1281. doi: 10.1016/j.anihpc.2014.09.003

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