We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case. INTRODUCTION Existence and non-existence of solutions of semilinear elliptic srstems has been a subject of active research recently; see for example [FF], [L3], [PS]. Such systems are called variational if solutions can be viewed as critical points of an associated functional defined on a suitable function space. Restricting our attention to systems with two unknowns, we distinguish two classes of variational elliptic systems: (a) potential systems; (b) Hamiltonian systems. Potential systems are of the form (A) -Llu = ,,> (u, v), ztL2v = ,,> (u, v), where L1 and L2 are second order selfadjoint elliptic operators. Formally these are the Euler-Lagrange equations of the functional f (u, v) = (L1u, u) i (L2v, v)-tH(u, v) . Here (, ) is the L2 inner product for functions defined on the underlying domain, and tt(u, v) = S H(u(x), v(x))dx. The functional f is often called the Lagrangian. Its critical points correspond to weak soltbtions of (A). The appropr:;ate choice of the function spaces for u and v follows by requiring that the quadratic part of f be well defined. This naturally leads to Sobolev spaces with square irltegrable first derivatives. Since also A(, ) must be well defined, by virtue of the embedding theorems, some growth restrictions on this latter function are required. An extra difficulty appears if we consider system (A) with a plus sign in the second equation, for then the corresponding functiorl f has a strongly indefinite quadratic part. For systems of superlinear type and with sllbcritical growth the Mountain Pass Theorem [AR] or, in the strongly indefinite case, the Benci-Rabinowitz Theorem Received by the editors June 5, 1996. 1991 Mathematics Subject Classification. Primary 35J50, 35J55, 35J65.