  In the present paper, we deal with a quasilinear elliptic equation involving
a critical Sobolev exponent on non-compact Randers spaces. Under very general
assumptions on the perturbation, we prove the existence of a non-trivial
solution. The approach is based on the direct methods of the calculus of
variations. One of the key steps is to prove that the energy functional
associated with the problem is weakly lower semicontinuous on small balls of
the Sobolev space, which is provided by a general inequality.
