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

Abstract

We consider the nonlinear Sturm-Liouville differential operator F (u) = −u ′′ + f (u) for u ∈ H 2 D ([0, π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity f : R → R 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. IntroductionConsider the nonlinear Sturm-Liouville problem −u ′′ (t) + f (u(t)) = g(t), u(0) = u(π) = 0 and for any smooth nonlinearity f : R → R denote by F the differential operatoris the Sobolev space of functions u(t) with square integrable second derivatives which satisfy Dirichlet boundary conditions u(0) = u(π) = 0 2000 Mathematics Subject Classification. Primary 34L30, 58B05, Secondary 34B15, 46T05. Keywords and phrases Sturm-Liouville, nonlinear differential operators, infinite dimensional manifolds.D. Burghelea was supported by an NSF grant. N. Saldanha and C. Tomei acknowledge the support of CNPq, Faperj and Finep.

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“…We begin with an easy consequence of item 1 of Proposition 3.1 in [7]. We now use this lemma to prove an amplification (the lemma is the degenerate case K = ∅).…”

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