  We present a new construction of the Euclidean $\Phi^4$ quantum field theory
on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an
approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined
on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We
introduce a new renormalized energy method in weighted spaces and prove
tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$,
$M \rightarrow \infty$. Every limit point is non-Gaussian and satisfies
reflection positivity, translation invariance and stretched exponential
integrability. These properties allow to verify the Osterwalder--Schrader
axioms for a Euclidean QFT apart from rotation invariance and clustering. Our
argument applies to arbitrary positive coupling constant, to multicomponent
models with $O(N)$ symmetry and to some long-range variants. Moreover, we
establish an integration by parts formula leading to the hierarchy of
Dyson--Schwinger equations for the Euclidean correlation functions. To this
end, we identify the renormalized cubic term as a \emph{distribution} on the
space of Euclidean fields.
