scholarly journals Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

2020 ◽  
Vol 19 (2) ◽  
pp. 1111-1128
Author(s):  
Xiaoxue Ji ◽  
◽  
Pengcheng Niu ◽  
Pengyan Wang
Keyword(s):  
Direct Method ◽  
Existence Results ◽  
Fractional System ◽  
Moving Spheres

Liouville theorem for fractional Hénon–Lane–Emden systems on a half space

2019 ◽  
Vol 150 (6) ◽  
pp. 3060-3073
Author(s):  
Phuong Le
Keyword(s):  
Strong Solution ◽  
Half Space ◽  
Liouville Theorem ◽  
Direct Method ◽  
Fractional Systems ◽  
Fractional System ◽  
Beta 2 ◽  
Image Position ◽  
A Direct Method

AbstractThis paper is concerned with the fractional system \begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}where n ⩾ 2, 0 < α, β < 2, a > −α, b > −β and p, q ⩾ 1. By exploiting a direct method of scaling spheres for fractional systems, we prove that if $p \leqslant \frac {n+\alpha +2a}{n-\beta }$, $q \leqslant \frac {n+\beta +2b}{n-\alpha }$, $p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$ and (u, v) is a nonnegative strong solution of the system, then u ≡ v ≡ 0.

scholarly journals Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunting Li ◽  
Yaqiong Liu ◽  
Yunhui Yi
Keyword(s):  
Linear System ◽  
Fractional Laplacian ◽  
Critical Case ◽  
Direct Method ◽  
Maxwell System ◽  
Nonlocal Nonlinearity ◽  
Nonnegative Solutions ◽  
Moving Spheres ◽  
Funct Anal

AbstractThis paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , where $0<\alpha \leq 2$ 0 < α ≤ 2 , $n\geq 2$ n ≥ 2 , $0<\sigma <n$ 0 < σ < n , and $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ 0 < p 1 , q 1 ≤ 2 n − σ n − α , $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ 0 < p 2 , q 2 ≤ n + α − σ n − α . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution $(u,v)$ ( u , v ) in the critical case and nonexistence of positive solutions in the subcritical cases.

scholarly journals A direct method of moving spheres on fractional order equations

2017 ◽  
Vol 272 (10) ◽  
pp. 4131-4157 ◽  
Author(s):  
Wenxiong Chen ◽  
Yan Li ◽  
Ruobing Zhang
Keyword(s):  
Fractional Order ◽  
Direct Method ◽  
Moving Spheres ◽  
A Direct Method

scholarly journals A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
K. Kavitha ◽  
Kottakkaran Sooppy Nisar ◽  
Anurag Shukla ◽  
Velusamy Vijayakumar ◽  
Shahram Rezapour
Keyword(s):  
Fixed Point ◽  
Nonlocal Condition ◽  
Infinite Delay ◽  
Existence Results ◽  
Sobolev Type ◽  
Differential Systems ◽  
Fractional Delay ◽  
Fractional System ◽  
Fixed Point Technique

AbstractThe goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch’s fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included.

scholarly journals Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

2017 ◽  
Vol 71 (2) ◽  
pp. 1
Author(s):  
Sonia Acinas ◽  
Fernando Mazzone
Keyword(s):  
Sobolev Space ◽  
Periodic Solutions ◽  
Potential Function ◽  
Calculus Of Variations ◽  
Hamiltonian Systems ◽  
Direct Method ◽  
Existence Results ◽  
Lagrange Equations ◽  
Space Setting

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space \(W^1L^\Phi([0,T])\). We employ the direct method of calculus of variations and we consider  a potential  function \(F\) satisfying the inequality \(|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)\), with \(b_1, b_2\in L^1\) and  certain \(N\)-functions \(\Phi_0\).

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.

scholarly journals Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jingwei Deng ◽  
Weiyuan Ma ◽  
Kaiying Deng ◽  
Yingxing Li
Keyword(s):  
Dynamical Systems ◽  
Fractional Calculus ◽  
Comparison Principle ◽  
Direct Method ◽  
Physical Space ◽  
Central Issue ◽  
Lyapunov Direct Method ◽  
Fractional Systems ◽  
Fractional System ◽  
Theoretical Results

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. This paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.

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