  We study $H^1$ versus $C^1$ local minimizers for functionals defined on
spaces of symmetric functions, namely functions that are invariant by the
action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases,
are associated with some elliptic partial differential equations that may have
supercritical growth. So we also prove some results on classical regularity for
symmetric weak solutions for a general class of semilinear elliptic equations
with possibly supercritical growth. We then apply these results to prove the
existence of a large number of classical positive symmetric solutions to some
concave-convex elliptic equations of H\'enon type.
