scholarly journals A Class of Variable-Order Fractional p · -Kirchhoff-Type Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yong Wu ◽  
Zhenhua Qiao ◽  
Mohamed Karim Hamdani ◽  
Bingyu Kou ◽  
Libo Yang
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Variational Method ◽  
Elliptic System ◽  
Type Systems ◽  
Ekeland Variational Principle ◽  
Variable Order ◽  
Variable Exponents ◽  
Kirchhoff Type ◽  
Direct Variational Method

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional p x -operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

scholarly journals ON AN ELLIPTIC SYSTEM OF P(X)-KIRCHHOFF-TYPE UNDER NEUMANN BOUNDARY CONDITION

2012 ◽  
Vol 17 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Zehra Yucedag ◽  
Mustafa Avci ◽  
Rabil Mashiyev
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Principle ◽  
Variational Method ◽  
Elliptic System ◽  
Neumann Boundary Condition ◽  
Existence Of Solutions ◽  
Neumann Boundary ◽  
Kirchhoff Type ◽  
Direct Variational Method

In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution.

Existence of a solution for a nonlocal elliptic system of (p(x),q(x))-Kirchhoff type

2018 ◽  
Vol 9 (3) ◽  
pp. 221-233
Author(s):  
Ali Taghavi ◽  
Horieh Ghorbani
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Variational Methods ◽  
Elliptic System ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Ekeland's Variational Principle ◽  
Existence Of A Solution ◽  
Kirchhoff Type

Abstract In this paper, we consider the system \left\{\begin{aligned} &\displaystyle{-}M_{1}\bigg{(}\int_{\Omega}\frac{\lvert% \nabla u\rvert^{p(x)}+\lvert u\rvert^{p(x)}}{p(x)}\,dx\biggr{)}(\Delta_{p(x)}u% -\lvert u\rvert^{p(x)-2}u)=\lambda a(x)\lvert u\rvert^{r_{1}(x)-2}u-\mu b(x)% \lvert u\rvert^{\alpha(x)-2}u&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M_{2}\bigg{(}\int_{\Omega}\frac{\lvert\nabla v\rvert^{q(x)}+% \lvert v\rvert^{q(x)}}{q(x)}\,dx\biggr{)}(\Delta_{q(x)}v-\lvert v\rvert^{q(x)-% 2}v)=\lambda c(x)\lvert v\rvert^{r_{2}(x)-2}v-\mu d(x)\lvert v\rvert^{\beta(x)% -2}v&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω is a bounded domain in {\mathbb{R}^{N}} ( {N\geq 2} ) with a smooth boundary {\partial\Omega} , {M_{1}(t),M_{2}(t)} are continuous functions and {\lambda,\mu>0} . We prove that for any {\mu>0} there exists {\lambda_{*}} sufficiently small such that for any {\lambda\in(0,\lambda_{*})} the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.

scholarly journals Existence of solutions for an elliptic p(x)-Kirchhoff-type systems in unbounded domain

2018 ◽  
Vol 36 (3) ◽  
pp. 193-205 ◽  
Author(s):  
Abdelamlek Brahim ◽  
Ali Djellit ◽  
Saadia Tas
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Unbounded Domain ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Main Tool ◽  
Type Systems ◽  
Mountain Pass ◽  
Nonlocal Term ◽  
Kirchhoff Type

In this paper we study of the existence of solutions for a class of elliptic system with nonlocal term in R^{N}. The main tool used is the variational method, more precisely, the Mountain Pass Theorem.

scholarly journals Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Yating Guo ◽  
Guoju Ye
Keyword(s):  
Variational Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Variable Order ◽  
Variable Exponents ◽  
Type Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Kirchhoff Type Problem ◽  
Laplacian Equations

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

scholarly journals Nontrivial Solutions for a Class of p-Kirchhoff Dirichlet Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xian Hu ◽  
Yong-Yi Lan
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Variational Method ◽  
Critical Point ◽  
Critical Point Theory ◽  
Dirichlet Boundary ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Nontrivial Solutions ◽  
Kirchhoff Type

This paper is devoted to the following p-Kirchhoff type of problems −a+b∫Ω∇updxΔpu=fx,u,x∈Ωu=0,x∈∂Ω with the Dirichlet boundary value. We show that the p-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.

HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

Ferroelectricity enhancement in confined nanorods: Direct variational method

2006 ◽  
Vol 73 (21) ◽  
Author(s):  
Anna N. Morozovska ◽  
Eugene A. Eliseev ◽  
Maya D. Glinchuk
Keyword(s):  
Variational Method ◽  
Direct Variational Method
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