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 the coupled quasilinear elliptic system of gradient type{- div(A(x, u)vertical bar del u vertical bar(p1-2) del u) + 1/p(1) A(u)(x, u)vertical bar del u vertical bar(p1) = G(u)(x, u, v) in Omega,- div(B( x, v)vertical bar del v vertical bar(p2-2)del v) + 1/p(2) B-v(x, v)vertical bar del v vertical bar(p2) = G(v)(x, u, v) in Omega,u = v = 0 on partial derivative Omega,where Omega subset of R-N is an open bounded domain, p(1), p(2) > 1 and A(x, u), B(x, v) are C-1-Caratheodory functions on Omega x R with partial derivatives A(u)(x, u), respectively B-v(x, v), while G(u)(x, u, v), G(v)(x, u, v) are given Caratheodory maps defined on Omega x R x R which are partial derivatives of a function G( x, u, v). We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional J, related to problem (P), admits at least one critical point in the "right" Banach space X. Moreover, if J is even, then (P) has infinitely many weak bounded solutions. 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.