Fredholm properties and nonlineare Dirichlet problems for mixed type operators

For a class of linear partial differential operators of mixed elliptic-hyperbolic type with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent spectral theory of [Lupo, Monticelli, Payne 2012] to establish a Fredholm alternative for weak solutions of the linear Dirichlet problem. This alternative is then used to study nonlinear Dirichlet problems with at most asymptotically linear nonlinearities, both in resonant and nonresonant cases. In particular, we obtain solvability results in nonresonant situations, a nonlinear Fredholm alternative (in the spirit of Landesman and Lazer) valid in both nonresonant and strongly resonant situations and establish a multiplicity result valid in nonresonant and weakly resonant situations.