Sharp Liouville Theorems

Abstract

Consider the equation div⁡(φ2⁢∇⁡σ)=0{\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝN{\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 ∫BR(φ⁢σ)2≤C⁢R2\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 01{R>1} and Ψ′>0{\Psi^{\prime}>0}, this condition is equivalent to ∫1∞1Ψ′=∞.\int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.

Keywords

• liouville theorems
• de giorgi’s conjecture
• 35b08 35b35 35b53 35j91