Global compactness results for singular quasilinear elliptic problems with critical Sobolev exponents and applications

In this paper, we give the complete descriptions for the PS-sequences of the energy functionals associated with the singular quasilinear elliptic problems involving critical Sobolev exponents on a bounded smooth domain invariant under the action of a group of orthogonal transformations and the whole space, respectively. As applications, we prove the existence of positive and nodal solutions for the singular quasilinear elliptic problem involving critical Sobolev exponents.