In the context of a variational model for the epitaxial growth of strained elastic films,
we study the effects of the presence of anisotropic surface energies in the determination
of equilibrium configurations. We show that the threshold effect that describes the
stability of flat morphologies in the isotropic case remains valid for weak anisotropies,
but is no longer present in the case of highly anisotropic surface energies, where we show
that the flat configuration is always a local minimizer of the total energy. Following the
approach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained
elastic films: second order minimality conditions and qualitative properties of solutions.
Preprint], we obtain these results by means of a minimality criterion based on the
positivity of the second variation.