  Using variational methods, we obtain several multiplicity results for double
phase problems that involve variable exponents and a new type of critical
growth. This new critical growth is better suited for double phase problems
when compared to previous works on the subject. In order to overcome the lack
of compactness caused by the critical exponents, we establish a
concentration-compactness principle of Lions type for spaces associated with
double phase operators, which is of independent interest to us. Our results are
new, even in the case of constant exponents.
