scholarly journals Nontrivial Solutions of the Kirchhoff-Type Fractional p-Laplacian Dirichlet Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Taiyong Chen ◽  
Wenbin Liu ◽  
Hua Jin
Keyword(s):  
Dirichlet Problem ◽  
Fractional Derivative ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Fractional Derivative Operators

In this article, we consider the new results for the Kirchhoff-type p-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.

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.

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

scholarly journals Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 83 ◽  
Author(s):  
Mohammed Massar ◽  
EL Miloud Hssini ◽  
Najib Tsouli
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Saddle Point Theorem ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

scholarly journals Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators

2021 ◽  
Vol 45 (5) ◽  
pp. 797-813
Author(s):  
SAJID IQBAL ◽  
◽  
GHULAM FARID ◽  
JOSIP PEČARIĆ ◽  
ARTION KASHURI
Keyword(s):  
Fractional Derivative ◽  
Convex Functions ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Hardy Type Inequalities ◽  
Exponential Convexity ◽  
Hardy Type ◽  
Liouville Fractional Derivative ◽  
Fractional Derivative Operators

In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.

Infinitely many Solutions of Semi-Linear Elliptic Equation with a Logarithmic Nonlinear Term

2014 ◽  
Vol 496-500 ◽  
pp. 2216-2219
Author(s):  
Yuan Li ◽  
Jiang Qin
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Weak Solutions ◽  
Sobolev Inequality ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Logarithmic Sobolev Inequality ◽  
Infinitely Many Solutions ◽  
Linear Elliptic Equation ◽  
Perturbation Theorem

The semi-linear elliptic equation is an important model in Mathematic, Physics. In this paper, we study the Dirichlet problem of semi-linear elliptic equation with a logarithmic nonlinear term. By using the logarithmic Sobolev inequality, mountain pass theorem and perturbation theorem, we obtain infinitely many nontrivial weak solutions, and also the energy of the solution is positive.

scholarly journals "A study of existence and multiplicity of positive solutions for nonlinear fractional differential equations with nonlocal boundary conditions"

2021 ◽  
Vol 66 (2) ◽  
pp. 361-380
Author(s):  
Noureddine Bouteraa ◽  
Slimane Benaicha
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Index Theory ◽  
Nonlocal Boundary Conditions ◽  
Point Theory ◽  
Integral Boundary ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Multiplicity Of Positive Solutions

"This paper deals with the existence, uniqueness and the multiplicity of solutions for a class of fractional di erential equations boundary value prob- lems involving three-point nonlocal Riemann-Liouville fractional derivative and integral boundary conditions. Our results are based on some well-known tools of xed point theory such as Banach contraction principle, xed point index theory and the Leggett-Williams xed point theorem. As applications, some examples are presented at the end to illustrate the main results."

Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Moloud Makvand Chaharlang ◽  
Abdolrahman Razani
Keyword(s):  
Boundary Condition ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Mountain Pass Theorem ◽  
Minimum Principle ◽  
Neumann Boundary ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

AbstractIn this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qiongfen Zhang
Keyword(s):  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Impulsive Effects ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Vibration Problems ◽  
Damped Vibration Problems

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains

2016 ◽  
Vol 146 (2) ◽  
pp. 435-448 ◽  
Author(s):  
Juntao Sun ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Bounded Domains ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

scholarly journals Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuping Cao ◽  
Chuanzhi Bai
Keyword(s):  
Dirichlet Boundary ◽  
Main Tool ◽  
Dirichlet Boundary Conditions ◽  
Type Problem ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Integrodifferential Operators ◽  
Homogeneous Dirichlet Boundary Conditions

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.

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