Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type

The aim of this paper is investigating the existence of one or more weak solutions of a family of coupled gradient-type quasilinear elliptic system of (p,q)-Laplacian type defined on an open bounded domain. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses such a problem has a variational structure and the related functional J admits at least one critical point in the ``right'' Banach space X. Moreover, if J is even, then infinitely many weak bounded solutions exist. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a "good" decomposition of the Banach space X and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.