On generalized Moser-Trudinger inequalities without boundary condition

Abstract

We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality.

As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.

Keywords

• Orlicz space
• Orlicz-Sobolev space
• embedding theorem
• sharp constants
• Moser-Trudinger inequality
• concentration-compactness principle