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

Abstract

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.