  We obtain critical embeddings and the concentration-compactness principle for
the anisotropic variable exponent Sobolev spaces. As an application of these
results,we confirm the existence of and find infinitely many nontrivial
solutions for a class of nonlinear critical anisotropic elliptic equations
involving variable exponents and two real parameters. With the groundwork laid
in this work, there is potential for future extensions, particularly in
extending the concentration-compactness principle to anisotropic fractional
order Sobolev spaces with variable exponents in bounded domains. This extension
could find applications in solving the generalized fractional Brezis-Nirenberg
problem.
