  Within mean-field theory we study wetting of elastic substrates. Our analysis
is based on a grand canonical free energy functional of the fluid number
density and of the substrate displacement field. The substrate is described in
terms of the linear theory of elasticity, parametrized by two Lam\'e
coefficients. The fluid contribution is of the van der Waals type. Two
potentials characterize the interparticle interactions in the system. The
long-ranged attraction between the fluid particles is described by a potential
$w(r)$, and $v(r)$ characterizes the substrate-fluid interaction. By
integrating out the elastic degrees of freedom we obtain an effective theory
for the fluid number density alone. Its structure is similar to the one for
wetting of an inert substrate. However, the potential $w(r)$ is replaced by an
effective potential which, in addition to $w(r)$, contains a term bilinear in
$v(r)$. We discuss the corresponding wetting transitions in terms of an
effective interface potential $\omega(\ell)$, where $\ell$ denotes the
thickness of the wetting layer. We show that in the case of algebraically
decaying interactions the elasticity of the substrate may suppress critical
wetting transitions, and may even turn them first order.
