scholarly journals Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations withp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Guowei Sun ◽  
Ali Mai
Keyword(s):  
Difference Equations ◽  
Critical Point Theory ◽  
Point Theory ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Superlinear Nonlinearity ◽  
Laplacian Equations

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discretep-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

scholarly journals Homoclinic Solutions for a Class of Nonlinear Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ali Mai ◽  
Zhan Zhou
Keyword(s):  
Difference Equations ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Superlinear Nonlinearity

We prove the existence of homoclinic solutions of a class of nonlinear difference equations with superlinear nonlinearity by using the generalized Nehari manifold approach. For the case where the nonlinearity is odd, we obtain infinitely many homoclinic solutions of the equations. Recent results in the literature are generalized and improved.

scholarly journals Regularity and multiplicity results for fractional (p,q)-Laplacian equations

2019 ◽  
Vol 22 (08) ◽  
pp. 1950065 ◽  
Author(s):  
Divya Goel ◽  
Deepak Kumar ◽  
K. Sreenadh
Keyword(s):  
Weak Solutions ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Subcritical Case ◽  
Existence Of A Solution ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold ◽  
Weak Harnack Inequality ◽  
Laplacian Equations

This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.

scholarly journals Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Defang Ma ◽  
Zhan Zhou
Keyword(s):  
Difference Equations ◽  
Mountain Pass Theorem ◽  
Sufficient Conditions ◽  
Homoclinic Solutions ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Special Cases ◽  
Discrete Nonlinear Schrödinger Equations ◽  
The Difference

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

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.

scholarly journals Sequences of positive homoclinic solutions to difference equations with variable exponent

2020 ◽  
Vol 70 (2) ◽  
pp. 417-430
Author(s):  
Robert Stegliński ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Variational Principle ◽  
Difference Equation ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Variable Exponent ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

Abstract We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with pk-Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

scholarly journals Existence and Multiplicity Results for A Fourth-Order Boundary Value Problem

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Nemat Nyamoradi ◽  
Mohamad Rasoul Hamidi
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Critical Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Nonsmooth Critical Point

Abstract In this paper we consider a class of a fourth-order boundary value problem. Using a variational method based on nonsmooth critical point theory, we prove the existence and multiplicity of solutions.

scholarly journals Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Jianmin Guo ◽  
Caixia Guo
Keyword(s):  
Positive Integer ◽  
Boundary Value Problem ◽  
Critical Point Theory ◽  
Difference Operator ◽  
Boundary Value ◽  
Point Theory ◽  
Point Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Forward Difference

By using Morse theory, the critical point theory, and the character of , we consider the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem subject to , where is a positive integer, is the forward difference operator defined by , and is continuous. In argument, Morse inequalities play an important role.

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