Embeddings of weighted Sobolev spaces and degenerate Dirichlet problems involving the weighted p-Laplacian

Abstract

We deal with two Dirichlet boundary value problems involving the weighted p-Laplacian. We introduce double weighted Sobolev spaces as solution spaces of the considered problems. We show the existence of weak solutions of these problems under an assumption of compact embedding involving the double weighted Sobolev spaces. Also, we establish sufficient conditions for fulfilling this embedding assumption.

Keywords:

• weighted Sobolev spaces
• Dirichlet spaces
• embedding properties
• quasiconformal mappings
• Dirichlet problems
• p-Laplacian

AMS Subject Classifications::

• 46E35
• 35J92
• 35J70
• 30C65

Acknowledgements

This work was partially supported by Israel Scientific Foundation (Grant No 1033/07). V.V. Motreanu was also supported by the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev.