Results on infinite-dimensional topology and applications to the structure of the critical set of nonlinear Sturm–Liouville operators

We consider the nonlinear Sturm–Liouville differential operator F(u)=−u″+f(u) for u HD2([0,π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity we show that there is a diffeomorphism in the domain of F converting the critical set C of F into a union of isolated parallel hyperplanes. For the proof, we show that the homotopy groups of connected components of C are trivial and prove results which permit to replace homotopy equivalences of systems of infinite-dimensional Hilbert manifolds by diffeomorphisms