Non-homogeneous Dirichlet boundary value problems in weighted Sobolev spaces

Abstract

The paper presents existence and multiplicity results for non-linear boundary value problems on possibly non-smooth and unbounded domains under possibly non-homogeneous Dirichlet boundary conditions. We develop here an appropriate functional setting based on weighted Sobolev spaces. Our results are obtained by using global minimization and a minimax approach using a non-smooth critical point theory.

Keywords:

• non-linear elliptic problem
• non-homogeneous Dirichlet boundary conditions
• weighted Sobolev spaces
• global optimization
• minimax principle

AMS Subject Classifications:

• 35J92
• 35J70
• 46E35
• 49J40

Acknowledgements

The authors are grateful to Lucas Fresse for his generous help throughout the preparation of this work.

Additional information

Funding

The third author is supported by a Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program [grant agreement number PIEF-GA-2010-274519].