scholarly journals A Liouville-Type Theorem for Elliptic Systems

1994 ◽  
pp. 405-415 ◽  
Author(s):  
D. G. de Figueiredo ◽  
P. L. Felmer
Keyword(s):  
Type Theorem ◽  
Elliptic Systems ◽  
Liouville Type ◽  
Liouville Type Theorem

scholarly journals A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

2020 ◽  
Vol 20 (2) ◽  
pp. 245-251
Author(s):  
Roberta Filippucci ◽  
Patrizia Pucci ◽  
Philippe Souplet
Keyword(s):  
Elliptic Equation ◽  
Harmonic Functions ◽  
Bounded Solution ◽  
Type Theorem ◽  
Elliptic Systems ◽  
Bounded Solutions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Superquadratic Growth ◽  
Gradient Bound

AbstractWe consider the elliptic equation {-\Delta u=u^{q}|\nabla u|^{p}} in {\mathbb{R}^{n}} for any {p>2} and {q>0}. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case {0<p<2} is considered. Some extensions to elliptic systems are also given.

scholarly journals A Liouville-type theorem for higher order elliptic systems

2014 ◽  
Vol 34 (9) ◽  
pp. 3317-3339 ◽  
Author(s):  
Frank Arthur ◽  
◽  
Xiaodong Yan ◽  
Mingfeng Zhao
Keyword(s):  
Type Theorem ◽  
Elliptic Systems ◽  
Higher Order ◽  
Liouville Type ◽  
Liouville Type Theorem

scholarly journals Liouville-type theorem for higher-order Hardy-Hénon system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kui Li ◽  
Zhitao Zhang
Keyword(s):  
Elliptic System ◽  
Type Theorem ◽  
Elliptic Systems ◽  
Higher Order ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Integral System ◽  
Moving Planes ◽  
Polyharmonic System ◽  
Moving Spheres

<p style='text-indent:20px;'>In this paper, we study higher-order Hardy-Hénon elliptic systems with weights. We first prove a new theorem on regularities of the positive solutions at the origin, then study equivalence between the higher-order Hardy-Hénon elliptic system and a proper integral system, and we obtain a new and interesting Liouville-type theorem by methods of moving planes and moving spheres for integral system. We also use this Liouville-type theorem to prove the Hénon-Lane-Emden conjecture for polyharmonic system under some conditions.</p>

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