Bifurcation without Fréchet differentiability at the trivial solution (Q2793946)

This paper is a survey of some interesting recent results, mostly obtained by the author (sometimes with his collaborator G. Evéquoz) on bifurcation for equations \(F(\mu,u)= 0\) in Banach spaces when the usual properties of Fréchet-differentiability for the linearized operator \(D(\mu, 0)\) do not hold and are replaced by Hadamard (or weak-Hadamard) differentiability. Several nonlinear problems, including critically tapered elastica and Schrödinger equations, served as motivation for this work.NEWLINENEWLINE Section 2 states the main notions of differentiability and bifurcation. Different difficulties arising in the generalization of known results in the classical case are analyzed and overcome in Section 3 using finite dimension reductions and the notions (new or not) of essential conditioning number, parity and Lipschitz modulus. Variational arguments in Hilbert spaces are also used in order to get bifurcation (and non-bifurcation) theorems. A simpler problem is treated in Hilbert spaces in Section 4. Finally, some second-order elliptic problems in \(\mathbb{R}^n\) are considered in Section 5, where bifurcation of bound states is studied.