Asymptotic bifurcation and second order elliptic equations on $R^N$

Abstract

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.