A simple proof of the optimal power in Liouville theorems

Abstract

Consider the equation div$({\phi }^2\nabla \sigma )=0$ in ${\mathbb{R}}^N,$ where $\phi >0$. It is well-known that if there exists $C>0$ such that ${\int }_{B_R}(\phi \sigma {)}^{2}dx\le C{R}^2$ for every $R\ge 1$ then $\sigma$ is necessarily constant. In this paper we prove that this result is not true if we replace $R^2$ by $R^k$ for $k>2$ in any dimension $N$. This question is related to a conjecture by De Giorgi.

9 pages. Version 2 contains new results concerning the Dirichlet and potential energies