Resonance with respect to the Fucik spectrum

Let L be a self-adjoint operator on L^2(Omega;R) with Omega a bounded and open subset of {R}^N. This article considers the resonance problem with respect to the Fucik spectrum of L, which means that we study equations of the form Lu = Alpha u^+ - Beta u^- + f(.,u), when the homogeneous equation Lu = Alpha u^+ - Beta u^- has non-trivial solutions. Using the computation of degrees that are not necessarily +1 or -1, we present results about the existence of solutions. Our results are illustrated with examples and can be seen as generalizations of Landesman-Lazer conditions. Non-existence results are also given.