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(ω_1 \nabla u) = ω_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $ω_1(x),ω_2(x)$. 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 $ω_1,ω_2$ near infinity. Also the monotonicity of $ω_1$ is involved in some results. Our methods here are the methods developed by Farina, \cite{f2}. We examine a specific class of weights $ω_1(x) = (|x|^2 +1)^\fracα{2}$ and $ω_2(x) = (|x|^2+1)^\fracβ{2} g(x)$ 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.

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