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Sharp Liouville Theorems

Consider the equation div$(φ^2 \nabla σ)=0$ in $\mathbb{R}^N,$ where $φ>0$. Berestycki, Caffarelli and Nirenberg proved that if there exists $C>0$ such that $\int_{B_R}(φσ)^2 \leq CR^2$ for every $R\geq 1$ then $σ$ is necessarily constant. In this paper we provide necessary and sufficient conditions on $0<Ψ\in C([1,\infty))$ for which this result remains true if we replace $R^2$ with $Ψ(R)$ in any dimension $N$. In the case of the convexity of $Ψ$ for large $R>1$ and $Ψ&#39;>0$, this condition is equivalent to $\displaystyle{\int_1^\infty\frac{1}{Ψ&#39;}=\infty}$.