Existence of solutions of quasilinear elliptic equations on manifolds with conic points

Abstract

Existence and regularity of solutions of quasilinear elliptic equations in nonsmooth domains have been interesting topics in the development of partial differential equations. The existence of finite-energy solutions of higher-order equations, also those with degenerations and singularities, can be shown by theories of monotone operators and topological methods. There are few results about singular solutions of second-order equations involving the p-Laplacian with the Dirac distribution on the right-hand side. So far the existence of singular solutions of higher-order equations with a prescribed asymptotic behavior has not been investigated. The aims of my dissertation are to look for finite-energy and singular solutions of quasilinear elliptic equations on manifolds with conic points. We single out realizations of the p-Laplacian in particular, (p>= 2), and a cone-degenerate operator in general, which are shown to belong to the class (S)_+. Assuming further coercivity conditions and employing mapping degree theory for generalized monotone mappings, we obtain existence for the prototypical example of the p-Laplacian and for general higher-order equations.