C¹(Rᴺ) Versus W¹,ᵖ(Rᴺ) Local Minimizers

Abstract

This paper deals with the energy functional associated with a quasilinear elliptic equation in R^N which is driven by the p-Laplacian operator. It is shown for such functional that any C¹(R^N) local minimizer in an appropriate sense is a W^1,p(R^N) local minimizer. This extends to R^N the celebrated property of Brezis-Nirenberg type known for bounded domains.