Asymptotic bifurcation and second order elliptic equations on ${ℝ}^{N}$
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1259-1281.

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation $G\left(u\right)=\lambda u$, where G acts between real Hilbert spaces and $\lambda \in ℝ$, 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.

DOI: 10.1016/j.anihpc.2014.09.003
Classification: 35J91,  47J15
Keywords: Asymptotic linearity, Asymptotic bifurcation, Nonlinear elliptic equation
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title = {Asymptotic bifurcation and second order elliptic equations on ${\mathbb{R}}^{N}$
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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, Volume 32 (2015) no. 6, pp. 1259-1281. doi : 10.1016/j.anihpc.2014.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.003/

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