Bifurcation from infinity for an asymptotically linear Schrödinger equation

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We consider an asymptotically linear Schrödinger equation -Δu + V(x)u = λu + f(x,u), x∈ Rⁿ, and show that if λ₀ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence (uₙ,λₙ) of solutions such that uₙ→∞ and λₙ→λ₀. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of λ₀ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.