An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

2020 ◽  
Vol 82 (1) ◽  
Author(s):  
Sheng Chen ◽  
Jie Shen
Keyword(s):  
Numerical Method ◽  
Fractional Laplacian ◽  
Laplacian Equation ◽  
Spectral Fractional Laplacian

Spectral fractional Laplacian problems

2016 ◽  
pp. 104-128
Author(s):  
Giovanni Molica Bisci ◽  
Vicentiu D. Radulescu ◽  
Raffaella Servadei
Keyword(s):  
Fractional Laplacian ◽  
Spectral Fractional Laplacian

Applications of monotone operators to a class of fractional Laplacian equation

2015 ◽  
Vol 26 (07) ◽  
pp. 1550043
Author(s):  
V. Raghavendra ◽  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Differential Operator ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Monotone Operators ◽  
Bounded Subset ◽  
Laplacian Equation ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Type

In this study we establish the existence of a weak solution for a class of nonlocal problem [Formula: see text] where [Formula: see text] is a general nonlocal integro-differential operator of fractional type, λ is a real parameter, Ω is an open bounded subset of ℝn(n > 2s, where s ∈(0, 1) is fixed) with continuous boundary ∂Ω. Here f, g1: Ω → ℝ and h : ℝ → ℝ are functions satisfying suitable hypotheses.

scholarly journals Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

2015 ◽  
Vol 4 (1) ◽  
pp. 37-58 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Existence Results ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Beta 2 ◽  
Fractional Laplace Operator

AbstractIn this article, we study the following p-fractional Laplacian equation: $ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

2016 ◽  
Vol 27 (05) ◽  
pp. 1650048 ◽  
Author(s):  
Li Ma
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Boundedness Of Solutions ◽  
Ginzburg Landau ◽  
Fractional Schrödinger Equation ◽  
Laplacian Equation ◽  
Kato Inequality ◽  
Fractional Schrodinger Equation

In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.

The maximum principles for fractional Laplacian equations and their applications

2017 ◽  
Vol 19 (06) ◽  
pp. 1750018 ◽  
Author(s):  
Tingzhi Cheng ◽  
Genggeng Huang ◽  
Congming Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Dirichlet Condition ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Left Half ◽  
Monotonicity Properties ◽  
Fractional Laplacian Operator ◽  
Laplacian Equations

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: [Formula: see text] Here [Formula: see text] is a domain (bounded or unbounded) in [Formula: see text] which is convex in [Formula: see text]-direction. [Formula: see text] is the nonlocal fractional Laplacian operator which is defined as [Formula: see text] Under various conditions on [Formula: see text] and on a solution [Formula: see text] it is shown that [Formula: see text] is strictly increasing in [Formula: see text] in the left half of [Formula: see text], or in [Formula: see text]. Symmetry (in [Formula: see text]) of some solutions is proved.

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