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.

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.

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