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 ${\omega }_1(x),{\omega }_2(x)$. 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.

arXiv commentTo appear in Proc. Amer. Math. Soc