  We are interested in the existence versus non-existence of non-trivial stable
sub- and super-solutions of {equation} \label{pop} -div(\omega_1 \nabla u) =
\omega_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth
weights $ \omega_1(x),\omega_2(x)$. We consider the cases $ f(u) = e^u, u^p$
where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence
results which depend on the dimension $N$ and also on $ p$ and the behaviour of
$ \omega_1,\omega_2$ near infinity. Also the monotonicity of $ \omega_1$ is
involved in some results. Our methods here are the methods developed by Farina,
\cite{f2}. We examine a specific class of weights $ \omega_1(x) = (|x|^2
+1)^\frac{\alpha}{2}$ and $ \omega_2(x) = (|x|^2+1)^\frac{\beta}{2} g(x)$ where
$ g(x)$ is a positive function with a finite limit at $ \infty$. For this class
of weights non-existence results are optimal. To show the optimality we use
various generalized Hardy inequalities.
