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

$-\operatorname {div}( \omega _1 \nabla u) = \omega _2 f(u)$

(1)

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. We examine a specific class of weights $\omega _1(x) = ( \vert x\vert^2 +1)^\frac {\alpha }{2}$ and $\omega _2(x) = ( \vert x\vert^2+1)^\frac { \beta }{2} g(x)$, 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.