Bifurcation from infinity for an asymptotically linear Schrödinger equation

Abstract

We consider an asymptotically linear Schrödinger equation $-\Delta u+V(x)u=\lambda u+f(x,u),x\in R^N$, and show that if ${\lambda }_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence $(u_n,{\lambda }_n)$ of solutions such that $\Vert u_n\Vert \to \infty$ and ${\lambda }_n\to {\lambda }_0$. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of ${\lambda }_0$ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.

19 pages