  We study a class of elliptic SPDEs with additive Gaussian noise on
$\mathbb{R}^2 \times M$, with $M$ a $d$-dimensional manifold equipped with a
positive Radon measure, and a real-valued non linearity given by the derivative
of a smooth potential $V$, convex at infinity and growing at most
exponentially. For quite general coefficients and a suitable regularity of the
noise we obtain, via the dimensional reduction principle discussed in our
previous paper on the topic, the identity between the law of the solution to
the SPDE evaluated at the origin with a Gibbs type measure on the abstract
Wiener space $L^2 (M)$. The results are then applied to the elliptic stochastic
quantization equation for the scalar field with polynomial interaction over
$\mathbb{T}^2$, and with exponential interaction over $\mathbb{R}^2$ (known
also as H{\o}eg-Krohn or Liouville model in the literature). In particular for
the exponential interaction case, the existence and uniqueness properties of
solutions to the elliptic equation over $\mathbb{R}^{2 + 2}$ is derived as well
as the dimensional reduction for the values of the ``charge parameter'' $\sigma
= \frac{\alpha}{2\sqrt{\pi}} < \sqrt{4 \left( 8 - 4 \sqrt{3} \right) \pi}
\simeq \sqrt{4.23\pi}$, for which the model has an Euclidean invariant
probability measure (hence also permitting to get the corresponding
relativistic invariant model on the two dimensional Minkowski space).
