Geometric aspects of Ambrosetti–Prodi operators with lipschitz nonlinearities

Abstract

Let the function u satisfy Dirichlet boundary conditions on a bounded domain Ω. What happens to the critical set of the Ambrosetti–Prodi operatorF(u) = −Δu−f(u). if the nonlinearity is only a Lipschitz map? It turns out that many properties which hold in the smooth case are preserved, despite of the fact that F is not even differentiable at some points. In particular, a global Lyapunov–Schmidt decomposition of great convenience for numerical solution of F(u) = g is still available.