On stable entire solutions of semi-linear elliptic equations with weights

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.