scholarly journals Bifurcation analysis for a modified quasilinear equation with negative exponent

2021 ◽  
Vol 11 (1) ◽  
pp. 684-701
Author(s):  
Siyu Chen ◽  
Carlos Alberto Santos ◽  
Minbo Yang ◽  
Jiazheng Zhou
Keyword(s):  
Bounded Domain ◽  
Bifurcation Analysis ◽  
Bifurcation Theory ◽  
Quasilinear Equation ◽  
Continuous Functions ◽  
Solution Curve ◽  
Bifurcation Parameter ◽  
Smooth Bounded Domain ◽  
Quasilinear Problem ◽  
Bounded Continuous Functions

Abstract In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

scholarly journals ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS WITH PARAMETERS

2010 ◽  
Vol 52 (2) ◽  
pp. 383-389 ◽  
Author(s):  
CHAOQUAN PENG
Keyword(s):  
Bounded Domain ◽  
Trivial Solution ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Asymptotically Linear ◽  
Smooth Bounded Domain ◽  
Negative Numbers ◽  
Solution Pair ◽  
Image Position

AbstractIn this paper, we show that the semi-linear elliptic systems of the form (0.1) possess at least one non-trivial solution pair (u, v) ∈ H01(Ω) × H01(Ω), where Ω is a smooth bounded domain in ℝN, λ and μ are non-negative numbers, f(x, t) and g(x, t) are continuous functions on Ω × ℝ and asymptotically linear at infinity.

scholarly journals Boundary Singularities of Solutions to Semilinear Fractional Equations

2018 ◽  
Vol 18 (2) ◽  
pp. 237-267 ◽  
Author(s):  
Phuoc-Tai Nguyen ◽  
Laurent Véron
Keyword(s):  
Large Class ◽  
Bounded Domain ◽  
Critical Exponents ◽  
Integral Form ◽  
Continuous Functions ◽  
Existence Of A Solution ◽  
Smooth Bounded Domain ◽  
Radon Measures ◽  
Fractional Equations ◽  
Boundary Trace

AbstractWe prove the existence of a solution of{(-\Delta)^{s}u+f(u)=0}in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functionsfsatisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where{f(u)=u^{p}}and μ is a Dirac mass, we show the existence of several critical exponentsp. We also demonstrate the existence of several types of separable solutions of the equation{(-\Delta)^{s}u+u^{p}=0}in{\mathbb{R}^{N}_{+}}.

Maximum Principles and ABP Estimates to Nonlocal Lane–Emden Systems and Some Consequences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Edir Junior Ferreira Leite
Keyword(s):  
Bounded Domain ◽  
Viscosity Solution ◽  
Lower Estimate ◽  
Continuous Functions ◽  
Nonlinear Perturbation ◽  
Maximum Principles ◽  
Laplace Operators ◽  
Smooth Bounded Domain ◽  
Principal Eigenvalues ◽  
Alpha Beta

Abstract This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s ⁢ u = λ ⁢ ρ ⁢ ( x ) ⁢ | v | α - 1 ⁢ v in  ⁢ Ω , ( - Δ ) t ⁢ v = μ ⁢ τ ⁢ ( x ) ⁢ | u | β - 1 ⁢ u in  ⁢ Ω , u = v = 0 in  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{n}\setminus\Omega,\end{aligned}\right. where s , t ∈ ( 0 , 1 ) {s,t\in(0,1)} , α , β > 0 {\alpha,\beta>0} satisfy α ⁢ β = 1 {\alpha\beta=1} , Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} , n ≥ 1 {n\geq 1} , and ρ and τ are continuous functions on Ω ¯ {\overline{\Omega}} and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small | Ω | {\lvert\Omega\rvert} has to be to ensure the positivity of the obtained solutions.

scholarly journals POSITIVE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS

2007 ◽  
Vol 49 (2) ◽  
pp. 377-390 ◽  
Author(s):  
CHAOQUAN PENG ◽  
JIANFU YANG
Keyword(s):  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Asymptotically Linear ◽  
Smooth Bounded Domain ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Solution Pair ◽  
Image Position

AbstractIn this paper, we show that the semilinear elliptic systems of the form (0.1) possess at least one positive solution pair (u, v) ∈ H10(Ω) × H10(Ω), where Ω is a smooth bounded domain in $\mathbb{R}^N$, f(x,t) and g(x, t) are continuous functions on $\Omega\times \mathbb{R}$ and asymptotically linear at infinity.

scholarly journals Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1 < p < 2), via Morse theory

2021 ◽  
pp. 2150065
Author(s):  
Giuseppina Vannella
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Quasilinear Equation ◽  
Multiple Positive Solutions ◽  
Type Problem ◽  
Poincaré Polynomial ◽  
Subcritical Growth ◽  
Quasilinear Problem

Let us consider the quasilinear problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] is a parameter and [Formula: see text] is a continuous function with [Formula: see text], having a subcritical growth. We prove that there exists [Formula: see text] such that, for every [Formula: see text], [Formula: see text] has at least [Formula: see text] solutions, possibly counted with their multiplicities, where [Formula: see text] is the Poincaré polynomial of [Formula: see text]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [Formula: see text], approximating [Formula: see text].

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

Application of Bifurcation Theory to Voltage Stability in Power System

2013 ◽  
Vol 811 ◽  
pp. 643-646
Author(s):  
Xue Song Zhou ◽  
Mo Chen ◽  
You Jie Ma
Keyword(s):  
Power System ◽  
Bifurcation Analysis ◽  
Bifurcation Point ◽  
Bifurcation Theory ◽  
Voltage Stability ◽  
Formal Definition ◽  
Voltage Stabilization ◽  
Advantages And Disadvantages ◽  
Voltage Instability ◽  
Dynamic Bifurcation

In order to study on the problem of voltage stability of power system, this paper describes the static bifurcation analysis and the dynamic bifurcation analysis in voltage stabilization analysis of power system and its relationship with the voltage stability,discusses the voltage instability caused by two main bifurcation formal definition, the occurrence of the conditions and the calculation of the bifurcation point, and points out advantages and disadvantages of various algorithms. Finally the paper looks forward to further study of the bifurcation theory in terms of voltage stability.

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