Bifurcation from infinity for an asymptotically linear Schrödinger equation

We consider an asymptotically linear Schrödinger equation $-Δu + V(x)u = λu + f(x,u), \ x\in R^N$, and show that if $λ_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence $(u_n,λ_n)$ of solutions such that $\|u_n\|\to\infty$ and $λ_n\toλ_0$. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of $λ_0$ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.

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