scholarly journals Symmetry and Nonexistence of Positive Solutions for Weighted HLS System of Integral Equations on a Half Space

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Linfen Cao ◽  
Zhaohui Dai
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Positive Solutions ◽  
Type Theorem ◽  
Integral Form ◽  
System Of Integral Equations ◽  
Method Of Moving Planes ◽  
Rotationally Symmetric ◽  
Moving Planes ◽  
Pohozaev Type Identity

We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric aboutxn-axis, which is much more general than the main result of Zhuo and Li, 2011.

scholarly journals Quantitative analysis of a system of integral equations with weight on the upper half space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sufang Tang ◽  
Jingbo Dou
Keyword(s):  
Growth Rate ◽  
Asymptotic Behavior ◽  
Quantitative Analysis ◽  
Decay Rate ◽  
Integral Equations ◽  
Half Space ◽  
Positive Solutions ◽  
Sobolev Inequality ◽  
System Of Integral Equations ◽  
Lagrange System

<p style='text-indent:20px;'>In this paper we analyzed the integrability and asymptotic behavior of the positive solutions to the Euler-Lagrange system associated with a class of weighted Hardy-Littlewood-Sobolev inequality on the upper half space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_+^n. $\end{document}</tex-math></inline-formula> We first obtained the optimal integrability for the solutions by the regularity lifting theorem. And then, with this integrability, we investigated the growth rate of the solutions around the origin and the decay rate near infinity.</p>

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.

scholarly journals Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Phuong Le ◽  
Hoang-Hung Vo
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Method Of Moving Planes ◽  
Quasilinear Elliptic Systems ◽  
Moving Planes ◽  
Quasilinear Elliptic

<p style='text-indent:20px;'>By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b20">20</xref>] to the system, in which substantial differences with the single cases are presented.</p>

scholarly journals Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

2022 ◽  
Author(s):  
Rong Zhang
Keyword(s):  
Positive Solution ◽  
Elliptic System ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Critical Case ◽  
Nonlinear Elliptic System ◽  
Subcritical Case ◽  
Nonlinear Elliptic ◽  
Method Of Moving Planes ◽  
Moving Planes

Abstract In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

scholarly journals A Sum Operator Method for the Existence and Uniqueness of Positive Solutions to a System of Nonlinear Fractional Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jing Wu ◽  
Tunhua Wu
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Iterative Scheme ◽  
Operator Method ◽  
Quadratic System ◽  
System Of Integral Equations ◽  
Fractional System

This paper is concerned with the existence and uniqueness of positive solutions for a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution to the system are established. Moreover, an iterative scheme is constructed for approximating the solution. The case of quadratic system of fractional integral equations is also considered.

Liouville type theorem for nonlinear elliptic equation involving Grushin operators

2015 ◽  
Vol 17 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Xiaohui Yu
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Type Theorem ◽  
Integral Form ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Main Technique ◽  
Grushin Operators ◽  
Moving Plane

In this paper, we study the nonexistence of positive solutions for the following elliptic equation [Formula: see text] where Lαu = Δxu + (α + 1)2|x|2αΔyu, α > 0, (x, y) ∈ ℝm × ℝk. We will prove that this problem possesses no positive solutions under some assumptions on the nonlinear term f. The main technique we use is the moving plane method in an integral form.

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.

scholarly journals One approach to the axisymmetric problem of impact of fine shells of the S.P. Timoshenko type on elastic half-space

2021 ◽  
pp. 77-89
Author(s):  
Vladislav Bogdanov
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Axisymmetric Problem ◽  
Infinite System ◽  
Elastic Half Space ◽  
Wave Processes ◽  
System Of Integral Equations ◽  
Timoshenko Type ◽  
Dynamics Problems ◽  
The Impact

Refined model of S.P. Timoshenko makes it possible to consider the shear and the inertia rotation of the transverse section of the shell. Disturbances spread in the shells of S.P. Timoshenko type with finite speed. Therefore, to study the dynamics of propagation of wave processes in the fine shells of S.P. Timoshenko type is an important aspect as well as it is important to investigate a wave processes of the impact, shock in elastic foundation in which a striker is penetrating. The method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind and the convergence of this solution are well studied. Such approach has been successfully used for cases of the investigation of problems of the impact a hard bodies and an elastic fine shells of the Kirchhoff-Love type on elastic a half-space and a layer. In this paper an attempt is made to solve the axisymmetric problem of the impact of an elastic fine spheric shell of the S.P. Timoshenko type on an elastic half-space using the method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind. It is shown that this approach is not acceptable for investigated in this paper axisymmetric problem. The discretization using the Gregory methods for numerical integration and Adams for solving the Cauchy problem of the reduced infinite system of Volterra equations of the second kind results in a poorly defined system of linear algebraic equations: as the size of reduction increases the determinant of such a system to aim at infinity. This technique does not allow to solve plane and axisymmetric problems of dynamics for fine shells of the S.P. Timoshenko type and elastic bodies. This shows the limitations of this approach and leads to the feasibility of developing other mathematical approaches and models. It should be noted that to calibrate the computational process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind.

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