scholarly journals Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Tao Sun ◽  
Hua Su
Keyword(s):  
Dirichlet Problem ◽  
Fractional Laplacian ◽  
Method Of Moving Planes ◽  
Moving Planes

In this paper, using the method of moving planes, we study the monotonicity in some directions and symmetry of the Dirichlet problem involving the fractional Laplacian − Δ α / 2 u x = f u x , x ∈ Ω , u x > 0 , x ∈ Ω , u x = 0 , x ∈ ℝ n \ Ω , in a slab-like domain Ω = ℝ n − 1 × 0 , h ⊂ ℝ n .

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.

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 < 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 < γ < 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.

Semilinear Dirichlet problem for the fractional Laplacian

2020 ◽  
Vol 193 ◽  
pp. 111512
Author(s):  
Krzysztof Bogdan ◽  
Sven Jarohs ◽  
Edyta Kania
Keyword(s):  
Dirichlet Problem ◽  
Fractional Laplacian
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