  This paper is concerned with a class of nonhomogeneous quasilinear elliptic
system driven by the locally symmetric potential and the small continuous
perturbations in the whole-space $\mathbb{R}^N$. By a variant of Clark's
theorem without the global symmetric condition and a Moser's iteration
technique, we obtain the existence of multiple solutions when the nonlinear
term satisfies some growth conditions only in a circle with center 0 and the
perturbation term is any continuous function with a small parameter and no any
growth hypothesis.
