Critical sets of nonlinear Sturm-Liouville operators of Ambrosetti-Prodi type

Abstract

The critical set C of the operator F:H^2_D([0,pi]) -> L^2([0,pi]) defined by F(u)=-u&#39;&#39;+f(u) is studied. Here X:=H^2_D([0,pi]) stands for the set of functions that satisfy the Dirichlet boundary conditions and whose derivatives are in L^2([0,pi]). For generic nonlinearities f, C=\cup C_k decomposes into manifolds of codimension 1 in X. If f&#39;&#39;<0 or f&#39;&#39;>0, the set C_j is shown to be non-empty if, and only if, -j^2 (the j-th eigenvalue of u -> u&#39;&#39;) is in the range of f&#39;. The critical components C_k are (topological) hyperplanes.