Eigenvalue, Unilateral Global Bifurcation and Constant Sign Solution for a Fractional Laplace Problem

2015 ◽  
Vol 25 (13) ◽  
pp. 1550183 ◽  
Author(s):  
Bian-Xia Yang ◽  
Hong-Rui Sun ◽  
Zhaosheng Feng
Keyword(s):  
Bounded Domain ◽  
Bifurcation Point ◽  
Principal Eigenvalue ◽  
Global Bifurcation ◽  
Constant Sign ◽  
Perturbation Function ◽  
Eigenvalues And Eigenfunctions ◽  
Laplace Problem ◽  
Fractional Laplace Operator ◽  
The Continuum

In this paper, we apply the Ljusternik–Schnirelmann theory to deal with the eigenvalues and eigenfunctions of the fractional Laplace operator. It shows that there exists a simple and isolated principal eigenvalue [Formula: see text] such that under certain conditions on the perturbation function [Formula: see text], [Formula: see text] is a bifurcation point of the problem [Formula: see text] where [Formula: see text], [Formula: see text] is a smooth and bounded domain, [Formula: see text] [Formula: see text] with [Formula: see text] and [Formula: see text] satisfies the Carathéodory condition in the first two variables. There are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text], consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of the unilateral global bifurcation result, we extend our investigation to the existence of constant sign solutions for a class of related nonlinear fractional Laplace problems. Some known results on the fractional Laplace problem in the literature are generalized.

Unilateral Global Bifurcation, Half-Linear Eigenvalues and Constant Sign Solutions for a Fractional Laplace Problem

2017 ◽  
Vol 27 (01) ◽  
pp. 1750015 ◽  
Author(s):  
Bian-Xia Yang ◽  
Hong-Rui Sun ◽  
Zhaosheng Feng
Keyword(s):  
Differential Equation ◽  
Global Bifurcation ◽  
Nonlinear Problems ◽  
Constant Sign ◽  
Bifurcation Structure ◽  
Laplace Operators ◽  
Constant Sign Solutions ◽  
Fractional Differential ◽  
Laplace Problem ◽  
The Continuum

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation [Formula: see text] with nondifferentiable nonlinearity [Formula: see text]. It shows that there are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text], and two unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of this unilateral global bifurcation results, we present the existence of the principal half-eigenvalues of the half-linear fractional eigenvalue problem. Finally, we deal with the existence of constant sign solutions for a class of fractional nonlinear problems. Main results of this paper generalize the known results on classical Laplace operators to fractional Laplace operators.

scholarly journals A result on the bifurcation from the principal eigenvalue of theAp-Laplacian

1997 ◽  
Vol 2 (3-4) ◽  
pp. 185-195 ◽  
Author(s):  
P. Drábek ◽  
A. Elkhalil ◽  
A. Touzani
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Bifurcation Point ◽  
Principal Eigenvalue ◽  
Bifurcation Problem

We study the following bifurcation problem in any bounded domainΩinℝN:{Apu:=−∑i,j=1N∂∂xi[(∑m,k=1Namk(x)∂u∂xm∂u∂xk)p−22aij(x)∂u∂xj]=                         λg(x)|u|p−2u+f(x,u,λ),u∈W01,p(Ω).. We prove that the principal eigenvalueλ1of the eigenvalue problem{Apu=λg(x)|u|p−2u,u∈W01,p(Ω),is a bifurcation point of the problem mentioned above.

Global “Eclipse” Bifurcation in a Twinkling Oscillator

2013 ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Bifurcation Point ◽  
High Frequency ◽  
Global Bifurcation ◽  
Low Frequency ◽  
Negative Stiffness ◽  
Symmetric System ◽  
Ambient Vibrations ◽  
High Frequency Oscillations ◽  
Mass Chain

This work regards the use of cubic springs with intervals of negative stiffness, in other words “snap-through” elements, in order to convert low-frequency ambient vibrations into high-frequency oscillations, referred to as “twinkling”. The focus of this paper is on a global bifurcation of a two-mass chain which, in the symmetric system, involves infinitely many equilibria at the bifurcation point. The structure of this “eclipse” bifurcation is uncovered, and perturbations of the bifurcation are studied. The energies associated with the equilibria are examined.

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

2019 ◽  
Vol 21 (01) ◽  
pp. 1750097
Author(s):  
Inbo Sim ◽  
Satoshi Tanaka
Keyword(s):  
Symmetry Breaking ◽  
Bifurcation Point ◽  
Global Bifurcation ◽  
One Dimensional ◽  
Bifurcation Points ◽  
Connected Set ◽  
Hénon Equation ◽  
The One ◽  
Symmetry Breaking Bifurcation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem [Formula: see text] where [Formula: see text] is a specified continuous parametrization function.

PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS

2020 ◽  
pp. 1-20 ◽  
Author(s):  
AZEB ALGHANEMI ◽  
HICHEM CHTIOUI
Keyword(s):  
Bounded Domain ◽  
Laplace Operator ◽  
Critical Points ◽  
Existence Theorems ◽  
Variational Problems ◽  
Bounded Domains ◽  
Critical Problem ◽  
Critical Points At Infinity ◽  
Fractional Laplace Operator

We consider the fractional critical problem $A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in $\unicode[STIX]{x1D6FA},u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , where $A_{s},s\in (0,1)$ , is the fractional Laplace operator and $K$ is a given function on a bounded domain $\unicode[STIX]{x1D6FA}$ of $\mathbb{R}^{n},n\geq 2$ . This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when $K$ is close to 1.

scholarly journals Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Benlong Xu
Keyword(s):  
Unit Ball ◽  
Linear Function ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Global Bifurcation ◽  
Asymptotically Linear ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Bifurcation Of Solutions

This paper mainly dealt with the exact number and global bifurcation of positive solutions for a class of semilinear elliptic equations with asymptotically linear function on a unit ball. As byproducts, some existence and multiplicity results are also obtained on a general bounded domain.

scholarly journals Continuum of one-sign solutions of one-dimensional Minkowski-curvature problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Yanqiong Lu ◽  
Zhengqi Jing
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Nonlinear Boundary Conditions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
Curvature Problem ◽  
The Continuum

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

scholarly journals The Principal Eigenvalue Problems for Perturbed Fractional Laplace Operators

2021 ◽  
Vol 52 ◽  
Author(s):  
Guangyu Zhao
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Laplace Operators ◽  
Weakly Coupled ◽  
Dynamics Of Population ◽  
Second Order Elliptic Operators ◽  
Fractional Laplace Operator

We study a variety of basic properties of the principal eigenvalue of a perturbed fractional Laplace operator and weakly coupled cooperative systems involving fractional Laplace operators. Our work extends a number of well-known properties regarding the principal eigenvalues of linear second-order elliptic operators with Dirichlet boundary condition to perturbed fractional Laplace operators. The establish results are also utilized to investigate the spatio-temporary dynamics of population models.

On the fractional Lazer-McKenna conjecture with critical growth

2021 ◽  
pp. 1-33
Author(s):  
Qi Li ◽  
Shuangjie Peng
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Reduction Method ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Positive Eigenfunction ◽  
Image Position ◽  
Fractional Laplace Operator

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$ . Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

Some global results for a class of homogeneous nonlocal eigenvalue problems

2019 ◽  
Vol 21 (03) ◽  
pp. 1750093 ◽  
Author(s):  
Guowei Dai
Keyword(s):  
Eigenvalue Problem ◽  
Bifurcation Point ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Positive Answer ◽  
Spectral Structure ◽  
Type Problem ◽  
Bifurcation Phenomenon ◽  
Kirchhoff Type Problem ◽  
Nonlocal Eigenvalue Problem

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

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