Structure of the Fucik spectrum and existence of solutions for equations with asymmetric nonlinearities

Let L : dom L subset of L-2(Ohm) --> L-2(Ohm) be a self-adjoint operator, Ohm being open and bounded in R-N. We give a description of the Fucik spectrum of L away from the essential spectrum. Let X be a point in the discrete spectrum of L; provided that some non-degeneracy conditions are satisfied, we prove that the Fucik spectrum consists locally of a finite number of curves crossing at (X, X). Each of these curves can be associated to a critical point of the function H : x --> < \x \ ,x > (L)2 restricted to the unit sphere in ker(L - lambdaI). The corresponding critical values determine the slopes of these curves. We also give global results describing the Fucik spectrum, and existence results for semilinear equations, by performing degree computations between the Fucik curves.