Sharp Liouville Theorems

Abstract

Abstract Consider the equation div ⁡ ( φ 2 ⁢ ∇ ⁡ σ ) = 0 {\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝ N {\mathbb{R}^{N}} , where φ > 0 {\varphi>0} . Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists C > 0 {C>0} such that ∫ B R ( φ ⁢ σ ) 2 ≤ C ⁢ R 2 \int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every R ≥ 1 {R\geq 1} , then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on 0 < Ψ ∈ C ⁢ ( [ 1 , ∞ ) ) {0<\Psi\in C([1,\infty))} for which this result remains true if we replace C ⁢ R 2 {CR^{2}} by Ψ ⁢ ( R ) {\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large R > 1 {R>1} and Ψ ′ > 0 {\Psi^{\prime}>0} , this condition is equivalent to ∫ 1 ∞ 1 Ψ ′ = ∞ . \int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.