Sharp Liouville Theorems

Abstract

Consider the equation div$({\phi }^2\nabla \sigma )=0$ in ${\mathbb{R}}^N,$ where $\phi >0$. Berestycki, Caffarelli and Nirenberg proved that if there exists $C>0$ such that ${\int }_{B_R}(\phi \sigma {)}^2\le C{R}^2$ for every $R\ge 1$ then $\sigma$ is necessarily constant. In this paper we provide necessary and sufficient conditions on $0<\Psi \in C([1,\infty ))$ for which this result remains true if we replace $R^2$ with $\Psi (R)$ in any dimension $N$. In the case of the convexity of $\Psi$ for large $R>1$ and ${\Psi }^{\prime }>0$, this condition is equivalent to ${\int }_1^{\infty }\frac{1}{{\Psi }^{\prime }}=\infty$.

13 pages