Sharp Liouville Theorems
| Autor: | Villegas Salvador |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Advanced Nonlinear Studies, Vol 21, Iss 1, Pp 95-105 (2021) |
| Druh dokumentu: | article |
| ISSN: | 1536-1365 2169-0375 |
| DOI: | 10.1515/ans-2020-2111 |
| Popis: | 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(φσ)2≤CR2\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. |
| Databáze: | Directory of Open Access Journals |
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