  We investigate stationary solutions of a thin-film model for liquid two-layer
flows in an energetic formulation that is motivated by its gradient flow
structure. The goal is to achieve a rigorous understanding of the contact-angle
conditions for such two-layer systems. We pursue this by investigating a
corresponding energy that favors the upper liquid to dewet from the lower
liquid substrate, leaving behind a layer of thickness $h_*$. After proving
existence of stationary solutions for the resulting system of thin-film
equations we focus on the limit $h_*\to 0$ via matched asymptotic analysis.
This yields a corresponding sharp-interface model and a matched asymptotic
solution that includes logarithmic switch-back terms. We compare this with
results obtained using $\Gamma$-convergence, where we establish existence and
uniqueness of energetic minimizers in that limit.
