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Abstract. We derive the asymptotic behaviour of the ground states of a system of two coupled semilinear Poisson equations with a strongly indefinite variational structure in the critical Sobolev growth case.

Abstract. We prove that for a class of infinite dimensional Hamiltonian systems certain homoclinic connections to the origin cease to exist when the nonlinearities have 'super-critical' growth. The proof is based on a variational principle and a Pohožaev type identity.

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This paper is concerned with the existence of periodic orbits on energy hypersurfaces in cotangent bundles of Riemannian manifolds defined by mechanical Hamiltonians. In [15] it was proved that, provided certain geometric assumptions are satisfied, regular mechanical hypersurfaces in R 2n , in particular non-compact ones, contain periodic orbits if one homology group among the top half does not vanish. In the present paper we extend the above mentioned existence result to a class of hypersurfaces in cotangent bundles of Riemannian manifolds with flat ends.

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