Nonlinear perturbations of some non-invertible differential operators (Q637040)

The paper deals with problems of the type \(L\,u=N_{\lambda }\,u,\) where \(L\) is a non-invertible differential operator having compact resolvent and nonnegative spectrum and \(N_\lambda \) is a nonlinear operator depending continuously on a small parameter \(\lambda ,\) \(N_0=0.\) Assuming the existence of a pair of (possibly non-ordered) lower and upper solutions belonging to the kernel of \(L,\) the authors prove the existence of a solution for \(\lambda \) sufficiently small. Applications are given for a wide class of problems, ranging from ODEs to equations of parabolic and elliptic types, and equations involving the \(p\)-Laplacian operator. Also, interesting non-existence results are given.