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.

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].

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

Multiplicity and concentration results for a class of critical fractional Schrödinger–Poisson systems via penalization method

2018 ◽  
Vol 22 (01) ◽  
pp. 1850078 ◽  
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Type System ◽  
Laplacian Operator ◽  
Positive Potential ◽  
Penalization Method ◽  
Subcritical Growth ◽  
Fractional Laplacian Operator ◽  
Poisson Type

We deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Poisson-type system with critical growth: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], is the fractional Laplacian operator, [Formula: see text] is a continuous positive potential and [Formula: see text] is a superlinear continuous function with subcritical growth. Using penalization techniques and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value.

scholarly journals Remarks on the Generalized Fractional Laplacian Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 320 ◽  
Author(s):  
Chenkuan Li ◽  
Changpin Li ◽  
Thomas Humphries ◽  
Hunter Plowman
Keyword(s):  
Differential Equations ◽  
Fractional Laplacian ◽  
Laplacian Operator ◽  
Generalized Convolution ◽  
Riesz Fractional Derivative ◽  
Derivative Operator ◽  
Delta Sequence ◽  
Remote Sites ◽  
Definition Of ◽  
Fractional Laplacian Operator

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [Formula: see text].

scholarly journals On some critical problems for the fractional Laplacian operator

2012 ◽  
Vol 252 (11) ◽  
pp. 6133-6162 ◽  
Author(s):  
B. Barrios ◽  
E. Colorado ◽  
A. de Pablo ◽  
U. Sánchez
Keyword(s):  
Fractional Laplacian ◽  
Laplacian Operator ◽  
Critical Problems ◽  
Fractional Laplacian Operator

The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 393-407 ◽  
Author(s):  
Wael Abdelhedi ◽  
Hichem Chtioui ◽  
Hichem Hajaiej
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Laplacian Operator ◽  
Fractional Laplacian Operator

AbstractWe study the following fractional Yamabe-type equation:\left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.Here Ω is a regular bounded domain of{\mathbb{R}^{n}},{n\geq 2}, and{A_{s}},{s\in(0,1)}, represents the fractional Laplacian operator{(-\Delta)^{s}}in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

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