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

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

2001-02-20

arxiv.org/abs/math/0102161v1

Mathematics

Functional Analysis

6 pages, no figures

Scientific paper

10.1088/0951-7715/15/4/306

The critical set C of the operator F:H^2_D([0,pi]) -> L^2([0,pi]) defined by F(u)=-u''+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''<0 or f''>0, the set C_j is shown to be non-empty if, and only if, -j^2 (the j-th eigenvalue of u -> u'') is in the range of f'. The critical components C_k are (topological) hyperplanes.

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