Nonhomogeneous quasilinear elliptic systems with small perturbations and lack of compactness

We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by Stuart [Milan J. Math. 79 (2011) 327–341] and depends not only on [Formula: see text] but also on u. We show the existence of multiple small solutions when the nonlinear term [Formula: see text] satisfies locally sublinear and symmetric conditions and the perturbation is any continuous function with a small coefficient and no growth hypothesis. Our technical approach is mainly based on a variant of Clark’s theorem without the global symmetric condition. We develop the Moser’s iteration technique to this quasilinear elliptic system with nonhomogeneous differential operators and obtain the relationship between [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text]. We overcome some difficulties which are caused by the nonhomogeneity of the differential operator and the lack of compactness of the Sobolev embedding.