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(2) −4φ+ αφ = φ. By this we mean that φ > 0 and that φ ∈ C2(R3). It is a classical fact (see Coffman [10]) that such solutions exist and are unique for the cubic nonlinearity. Moreover, they are radial and smooth. Similar facts are known for more general nonlinearities; see e.g., Strauss [45] and Berestycki and Lions [5] for existence and Kwon [30] for uniqueness in greater generality. Clearly, ψ = eitα 2 φ solves (1). We seek an H1-solution ψ of the form ψ = W +R where

Stable manifolds for an orbitally unstable nonlinear Schrödinger equation