Bifurcation from infinity for an asymptotically linear Schr\"odinger equation

We consider an asymptotically linear Schr"odinger equation $-\Delta u + V(x)u = \lambda u + f(x,u), \ x\in R^N$,andshowthatif$\lambda_0$ 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 $\mathrm{\parallel }{u}_n\mathrm{\parallel }\to \mathrm{\infty }$ 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.