scholarly journals Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains

1996 ◽  
Vol 15 (3) ◽  
pp. 619-635 ◽  
Author(s):  
Wolfgang Reichel
Keyword(s):  
Radial Symmetry ◽  
Exterior Domains ◽  
Fully Nonlinear ◽  
Overdetermined Problems

scholarly journals Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains

2021 ◽  
Vol 4 (6) ◽  
pp. 1-18
Author(s):  
Liliane Maia ◽  
◽  
Gabrielle Nornberg ◽  
Keyword(s):  
Dynamical System ◽  
Nonlinear Equations ◽  
Shooting Method ◽  
Radial Solutions ◽  
Exterior Domains ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Positive Radial Solutions

<abstract><p>In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.</p></abstract>

Positive solutions to nonlinear systems involving fully nonlinear fractional operators

2018 ◽  
Vol 21 (2) ◽  
pp. 552-574 ◽  
Author(s):  
Pengcheng Niu ◽  
Leyun Wu ◽  
Xiaoxue Ji
Keyword(s):  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Fractional Operators ◽  
Fully Nonlinear ◽  
Narrow Region ◽  
Method Of Moving Planes ◽  
Whole Space ◽  
Moving Planes ◽  
Fractional System

Abstract In this paper we consider the following fractional system $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy} . \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

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