scholarly journals Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

2012 ◽  
Vol 29 (5) ◽  
pp. 653-665 ◽  
Author(s):  
Guozhen Lu ◽  
Peiyong Wang ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Higher Order ◽  
Decay Estimates ◽  
Liouville Type ◽  
Liouville Type Theorems

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Decay Estimates ◽  
Fully Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

AbstractThe main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

scholarly journals A Liouville theorem for the higher-order fractional Laplacian

2019 ◽  
Vol 21 (02) ◽  
pp. 1850005 ◽  
Author(s):  
Ran Zhuo ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Liouville Theorem ◽  
Integral Form ◽  
Higher Order ◽  
Order Equation ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Higher Order Equation ◽  
Fractional Equation

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [Formula: see text] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [Formula: see text] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

Liouville-type results for fourth order elliptic equations

1986 ◽  
Vol 103 (3-4) ◽  
pp. 209-213 ◽  
Author(s):  
Vinod B. Goyal
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Liouville Type Results ◽  
Fourth Order Elliptic Equations

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

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.

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