scholarly journals Liouville type theorem and decay estimates for solutions of fully nonlinear elliptic equation

2013 ◽  
Vol 405 (2) ◽  
pp. 608-617
Author(s):  
Yongxia Hua ◽  
Xiaohui Yu
Keyword(s):  
Elliptic Equation ◽  
Nonlinear Elliptic Equation ◽  
Type Theorem ◽  
Decay Estimates ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic

A Liouville-type theorem for fully nonlinear CR invariant equations on the Heisenberg group

2021 ◽  
pp. 2150060
Author(s):  
Bo Wang
Keyword(s):  
Heisenberg Group ◽  
Viscosity Solutions ◽  
Comparison Principle ◽  
Type Theorem ◽  
Liouville Type ◽  
Nonlinear Operators ◽  
Liouville Type Theorem ◽  
Fully Nonlinear

We obtain a Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group.

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.

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