D1,2(RN) versus C(RN) local minimizer and a Hopf-type maximum principle

We consider functionals of the form Φ(u)=12∫RN|∇u|2−∫RNb(x)G(u) on D1,2(RN), N≥3, whose critical points are the weak solutions of a corresponding elliptic equation in the whole RN. We present a Brezis-Nirenberg type result and a Hopf-type maximum principle in the context of the space D1,2(RN). More precisely, we prove that a local minimizer of Φ in the topology of the subspace V must be a local minimizer of Φ in the D1,2(RN)-topology, where V is given by V:={v∈D1,2(RN):v∈C(RN)withsupx∈RN⁡(1+|x|N−2)|v(x)|<∞}. It is well-known that the Brezis-Nirenberg result has been proved a strong tool in the study of multiple solutions for elliptic boundary value problems in bounded domains. We believe that the result obtained in this paper may play a similar role for elliptic problems in RN.