Generalized eigenvalue problem for totally discontinuous operators (Q977977)

The author studies the multivalued eigenvalue problems \(Au=\partial j(u)\) in the framework of Banach-Sobolev function spaces. Here \(A\) is a multivalued Leray-Lions operator and \(\partial j\) is a subdifferential (or a gradient in the sense of Clarke). Section 3 provides the necessary abstract results from critical point theory for multivalued operators with non-smooth functionals in Banach spaces allowing to apply a suitable version of the Mountain Pass Theorem. Section 4 gives examples of Banach function norms: weighted Orlicz-Sobolev spaces, Lorentz spaces, variable exponent Sobolev spaces with modular norms, etc. Dual spaces of Banach-Sobolev function spaces are also studied, and some computations of subdifferentials are provided. Last section develops some interesting examples of applications; the last one concerns the \(p\)-Laplacian operator with singular coefficients.