Bifurcation for some non-Fréchet differentiable problems

We study the bifurcation points of an equation of the form F(u)=λuF(u)=λu in a real Hilbert space. Since FF is only required to be Hadamard, but not Fréchet, differentiable at u=0u=0, bifurcation points need not belong to the spectrum of F′(0)F′(0). The abstract results are illustrated in the case of a nonlinear Schrödinger equation.