  In this work we consider a system of quasilinear elliptic equations driven by
an anisotropic $p$-Laplacian. The lower-order nonlinearities are in potential
form and exhibit critical Sobolev growth. We exhibit conditions on the
coefficients of the differential operator, the domain of the unknown function,
and the lower-order nonlinearities under which nontrivial solutions are
guaranteed to exist and conditions on these objects under which a nontrivial
solution does not exist.
