scholarly journals Radial symmetry of nonnegative solutions for nonlinear integral systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhenjie Li ◽  
Chunqin Zhou
Keyword(s):  
Singular Integral ◽  
Nonnegative Solution ◽  
Radial Symmetry ◽  
Monotonicity Condition ◽  
Independent Variables ◽  
Method Of Moving Planes ◽  
Integral Forms ◽  
Nonnegative Solutions ◽  
Integral Systems ◽  
Moving Planes

<p style='text-indent:20px;'>In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0&lt;\alpha&lt;n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;a_i/2&lt;\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ f_i(u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1\leq i\leq m $\end{document}</tex-math></inline-formula>, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables <inline-formula><tex-math id="M4">\begin{document}$ u_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ u_m $\end{document}</tex-math></inline-formula>. By the method of moving planes in integral forms, we show that the nonnegative solution <inline-formula><tex-math id="M8">\begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document}</tex-math></inline-formula> is radially symmetric when <inline-formula><tex-math id="M9">\begin{document}$ f_i $\end{document}</tex-math></inline-formula> satisfies some monotonicity condition.</p>

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

scholarly journals Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Linfen Cao ◽  
Xiaoshan Wang ◽  
Zhaohui Dai
Keyword(s):  
Nonlinear System ◽  
Unit Ball ◽  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Continuous Functions ◽  
Method Of Moving Planes ◽  
Moving Planes

In this paper, we study a nonlinear system involving the fractional p-Laplacian in a unit ball and establish the radial symmetry and monotonicity of its positive solutions. By using the direct method of moving planes, we prove the following result. For 0<s,t<1,p>0, if u and v satisfy the following nonlinear system -Δpsux=fvx;  -Δptvx=gux,  x∈B10;  ux,vx=0,  x∉B10. and f,g are nonnegative continuous functions satisfying the following: (i) f(r) and g(r) are increasing for r>0; (ii) f′(r)/rp-2, g′(r)/rp-2 are bounded near r=0. Then the positive solutions (u,v) must be radially symmetric and monotone decreasing about the origin.

Nonexistence results for a fractional Hénon–Lane–Emden equation on a half-space

2015 ◽  
Vol 26 (13) ◽  
pp. 1550110 ◽  
Author(s):  
Sufang Tang ◽  
Jingbo Dou
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Green Function ◽  
Euclidean Space ◽  
Dirichlet Condition ◽  
Emden Equation ◽  
Method Of Moving Planes ◽  
Integral Forms ◽  
Moving Planes ◽  
The Green Function

Consider the following Dirichlet problem involving the fractional Hénon–Lane–Emden Laplacian: [Formula: see text] where [Formula: see text] and [Formula: see text] is the upper half-Euclidean space. We first show that the above equation is equivalent to the following integral equation: [Formula: see text] where [Formula: see text] is the Green function in [Formula: see text] with the same Dirichlet condition. Then we prove the nonexistence of positive solutions by using the method of moving planes in integral forms.

scholarly journals Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

2021 ◽  
Vol 26 (2) ◽  
pp. 349-362
Author(s):  
Wenwen Hou ◽  
Lihong Zhang ◽  
Ravi P. Agarwal ◽  
Guotao Wang
Keyword(s):  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Laplacian Operator ◽  
Negative Power ◽  
Laplacian Equation ◽  
Narrow Region ◽  
Method Of Moving Planes ◽  
Moving Planes

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

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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