A simple proof of the optimal power in Liouville theorems

Abstract

Consider the equation $\text{div}({\phi }^2\nabla \sigma )=0$ in ${ℝ}^N$, where . It is well known [4, 2] that if there exists such that ${\displaystyle {\int }_{B_R}}{(\phi \sigma )}^{2}dx\le C{R}^2$ for every $R\ge 1$, then $\sigma$ is necessarily constant. In this paper we present a simple proof that this result is not true if we replace $R^2$ with $R^k$ for in any dimension $N$. This question is related to a conjecture by De Giorgi [7].