  We consider the Yamabe equation on a complete non-compact Riemannian manifold
and study the condition of stability of solutions. If $(M^m,g)$ is a closed
manifold of constant positive scalar curvature, which we normalize to be
$m(m-1)$, we consider the Riemannian product with the $n$-dimensional Euclidean
space: $(M^m \times \mathbf{R}^n, g+ g_E)$. And study the solution of the
Yamabe equation which depends only on the Euclidean factor. We show that there
exists a constant $\lambda (m,n)$ such that the solution is stable if and only
if $\lambda_1 \geq \lambda (m,n)$, where $\lambda_1$ is the first positive
eigenvalue of $-\Delta_g$. We compute $\lambda (m,n)$ numerically for small
values of $m,n$ showing in these cases that the Euclidean minimizer is stable
in the case $M=S^m $ with the metric of constant curvature. This implies that
the same is true for any closed manifold with a Yamabe metric.
