D^{1,2}(R^N) versus C(R^N) local minimizer and a Hopf-type Maximum Principle (with S. Carl and H. Tehrani)
Document Type
Article
Publication Date
4-25-2016
Publication Title
Journal of Differential Equations
Volume
261
Issue
3
First page number:
2006
Last page number:
2025
Abstract
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.
File Format
File Size
323.14 KB
Repository Citation
Carl, S.,
Costa, D. G.,
Tehrani, H.
(2016).
D^{1,2}(R^N) versus C(R^N) local minimizer and a Hopf-type Maximum Principle (with S. Carl and H. Tehrani).
Journal of Differential Equations, 261(3),
2006-2025.
http://dx.doi.org/10.1016/j.jde.2016.04.019
