© 2015 Elsevier Inc. We study H1 versus C1 local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of 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 Hénon type.
Iturriaga, L., Moreira dos Santos, E., & Ubilla, P. (2015). Local minimizers in spaces of symmetric functions and applications. Journal of Mathematical Analysis and Applications, 27-56. https://doi.org/10.1016/j.jmaa.2015.03.084