Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities

Abstract

In present work, we first establish the corresponding Sobolev inequality and Poincare inequality on the cone Sobolev spaces, and then, as an application of such inequalities, we prove the existence of non-trivial weak solution for Dirichlet boundary value problem for a class of non-linear elliptic equation on manifolds with conical singularities.