  In this paper we study the geometry of certain functionals associated to
quasilinear elliptic boundary value problems with a degenerate nonlocal term of
Kirchhoff type.
  Due to the degeneration of the nonlocal term it is not possible to directly
use classical results such as uniform a-priori estimates and "Sobolev versus
H\"older local minimizers" type of results. We prove that results similar to
these hold true or not, depending on how degenerate the problem is.
  We apply our findings in order to show existence and multiplicity of
solutions for the associated quasilinear equations, considering several
different interactions between the nonlocal term and the nonlinearity.
