  The semi-linear, elliptic PDE $AC_{\varepsilon}(u):=-\varepsilon^2\Delta
u+W'(u)=0$ is called the Allen-Cahn equation. In this article we will prove the
existence of finite energy solution to the Allen-Cahn equation on certain
complete, non-compact manifolds. More precisely, suppose $M^{n+1}$ (with
$n+1\geq 3$) is a complete Riemannian manifold of finite volume. Then there
exists $\varepsilon_0>0$, depending on the ambient Riemannian metric, such that
for all $0<\varepsilon\leq\varepsilon_0$, there exists
$\mathfrak{u}_{\varepsilon}:M\rightarrow (-1,1)$ satisfying
$AC_{\varepsilon}(\mathfrak{u}_{\varepsilon})=0$ with the energy
$E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty$ and the Morse index
$\text{Ind}(\mathfrak{u}_{\varepsilon})\leq 1$. Moreover,
$0<\liminf_{\varepsilon\rightarrow
0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})\leq\limsup_{\varepsilon\rightarrow
0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty.$ Our result is motivated
by the theorem of Chambers-Liokumovich and Song, which says that $M$ contains a
complete minimal hypersurface $\Sigma$ with $0<\mathcal{H}^n(\Sigma)<\infty.$
This theorem can be recovered from our result.
