A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications

Abstract

A generalization of the classical Caffarelli–Kohn–Nirenberg inequality is obtained in the setting of Orlicz–Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a nonhomogeneous and degenerate/singular elliptic PDE.

Keywords:

• Caffarelli–Kohn–Nirenberg inequality
• critical point
• degenerate elliptic equations
• Orlicz spaces

AMS Subject Classifications::

• 35D30
• 35J70
• 35J75
• 49J45

Acknowledgements

The research of Bocea was partially supported by the U.S. National Science Foundation under Grant No. DMS-0806789. Mihăilescu has been partially supported by the Grant CNCSIS PD-117/2010 ‘Probleme neliniare modelate de operatori diferenţiali neomogeni’.