Degenerate elliptic equations with singular nonlinearities

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of R^N, and compactness holds below a critical dimension N^#. The nonlinearity f (u) lies in a very general class and the results we present are new even for p=2. Due to the degeneracy of p-Laplace operator, for p\not=2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blow-up argument and stronger assumptions on the nonlinearity f (u) are required