The invertible, closed range, compact, Fredholm and isometric composition operators on Musielak-Orlicz spaces of Bochner type are characterized in the paper.
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.