Denotational semantics interpret the theories of various lambda calculi in various (set-theoretic, domain-theoretic, category-theoretic, game...) models. Let $T$ be the theory of one such lambda calculus $\lambda_?$. If I understand correctly, a denotational semantics for $\lambda_?$ is thus just a model for $T$.
The question here is, have applying the tools and methods of model theory to denotational semantics yielded useful results. For example, what does the Löwenheim–Skolem theorem tell us about the denotational semantics of various lambda calculi?