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In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, *i.e.*, the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

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