On differentiability and bifurcation

Abstract

For a function acting between Banach spaces, we recall the notions of Hadamard and w-Hadamard differentiability and their relation to the common notions of Gâteaux and Fréchet differentiability. We observe that even for a function F: H → H that is both Hadamard and w-Hadamard differentiable but not Fréchet differentiable at 0 on a real Hilbert space H, there may be bifurcation for the equation F(u) = λu at points λ which do not belong to the spectrum of F′(0). We establish some necessary conditions for λ to be a bifurcation point in such cases and we show how this result can be used in the context of partial differential equations such as

$$ - \Delta u\left( x \right) + q\left( x \right)u\left( x \right) = \lambda \left( {e^{\left| x \right|} u\left( x \right)} \right) for u \in H^2 \left( {\mathbb{R}^N } \right) $$

where this situation occurs.