scholarly journals Some Liouville theorems for Hénon type elliptic equations

2012 ◽  
Vol 262 (4) ◽  
pp. 1705-1727 ◽  
Author(s):  
Chao Wang ◽  
Dong Ye
Keyword(s):  
Elliptic Equations ◽  
Liouville Theorems

Liouville theorems for a class of fourth order elliptic equations

1980 ◽  
Vol 86 (1-2) ◽  
pp. 129-137 ◽  
Author(s):  
Vinod B. Goyal ◽  
Philip W. Schaefer
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Entire Solutions ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Liouville Type Theorems ◽  
Solutions Of Equations ◽  
Fourth Order Elliptic Equations

SynopsisLiouville type theorems are obtained for bounded entire solutions of equations of the form Δ2u − q(x)Δu + p(x)u = 0 by means of subharmonic functionals and Green type inequalities.

scholarly journals Sharp Liouville Theorems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Salvador Villegas
Keyword(s):  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Necessary And Sufficient Conditions ◽  
Qualitative Properties ◽  
Liouville Theorems ◽  
Necessary And Sufficient

AbstractConsider the equation {\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in {\mathbb{R}^{N}}, where {\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} such that \int_{B_{R}}(\varphi\sigma)^{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<\Psi\in C([1,\infty))} for which this result remains true if we replace {CR^{2}} by {\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large {R>1} and {\Psi^{\prime}>0}, this condition is equivalent to \int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.

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