SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.

Infinitely Many Solutions and a Ground-State Solution for Klein-Gordon Equation Coupled with Born-Infeld Theory

In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.

Multiplicity Results for Second Order Impulsive Differential Equations via Variational Methods

In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png" width="40" height="17" alt="" /> and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.

Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions

In this work,the solvability of a class of second-order Hamiltonian systems on time scales is generalized to non-local boundary conditions.The measurements obtained by non-local conditions are more accurate than those given by local conditions in some problems.Compared with the known results,this work establishes the variational structure in an appropriate Sobolev’s space.Then,by applying the mountain pass theorem and symmetric mountain pass theorem,the existence and multiplicity of the solutions are obtained.Finally,some examples with numerical simulation results are given to illustrate the correctness of the results obtained.

Nonsmooth critical point theory and applications to the spectral graph theory

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

EXISTENCE OF HOMOCLINIC ORBITS FOR A CLASS OF FIRST-ORDER DIFFERENTIAL DIFFERENCE EQUATIONS

In this article we consider via critical point theory the existence of homoclinic orbits of the first-order differential difference equation z（t）＋B（t）z（t）＋f（t,z（t＋τ）,z（t）,z（t-τ））=0.

Periodic Solutions for Some Second-order Differential System with p（t）-Laplacian

In this article, we investigate the existence of periodic solutions for a class of nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained by using critical point theory, which extend some known results.

Cheeger's cut, maxcut and the spectral theory of1-Laplacian on graphs

This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.

Critical Point Theorems and Applications to Differential Equations

This paper contains a generalization of the well–known Palais–Smale andCerami compactness conditions. The compactness condition introduced is used to prove some generalexistence theorems for critical points. Some applications are given to differential equations.

An Analytic Equation of State Based on SAFT-CP for Binary Non-Polar Alkane Mixtures Across the Critical Point

The description using an analytic equation of state of thermodynamic properties near the critical points of fluids and their mixtures remains a challenging problem in the area of chemical engineering. Based on the statistical associating fluid theory across the critical point (SAFT-CP), an analytic equation of state is established in this work for non-polar mixtures. With two binary parameters, this equation of state can be used to calculate not only vapor-liquid equilibria but also critical properties of binary non-polar alkane mixtures with acceptable deviations.

THE EXISTENCE OF HOMOCLINIC ORBITS FOR HAMILTONIAN SYSTEMS WITH THE POTENTIALS CHANGING SIGN

This paper studies the existence of nontrival homoclinic orbits of the Hamiltonian systems -L(t)q+V′(t,q)=0 by using the critical point theory, where the potential V(t,q)=b(t)W(q) can change sign. Under a new kind of "superquadratic" condition on W, some new results are obtained.

The Existence Theorems for a Class of Sublinear Elliptic Equations in R^N

An existence theorem for the solution to the equation -△u+b(x)u=f(x,u),in R^N is given by means of variational method where b(x)→∞,as丨x丨→∞ and f(x,s)has linear growth in s at infinity and sublinear growth in s at zero.For a special case,some multiplicity result is proved.

Periodic Solutions of Nonautonomous Second Order Hamiltonian Systems

In this paper, we develop the local linking theorem given by Li and Willein by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems （H） ü＋A（t）u＋∨V（t, u）=0, u∈R^N, t∈R. We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem.

Multiplicity results of breathers for the discrete nonlinear Schrodinger equations with unbounded potentials

We consider a class of discrete nonlinear Schrdinger equations with unbounded potentials. We obtain some new multiplicity results of breathers of the equations by using critical point theory. Our results greatly improve some recent results in the literature.

Existence and Concentration of Ground States of Coupled Nonlinear Schr■dinger Equations with Bounded Potentials

A 2-coupled nonlinear Schrbdinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for sufficiently small Planck constant is proved. As the Planck constant approaches zero, it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4.

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non–linear elliptic equations with Neumann boundary condition, and obtain at least seven non–trivial solutions in which two are positive, two are negative and three are sign–changing. The study of problem (1.1): is based on the variational methods and critical point theory. We form our conclusion by using the sub–sup solution method, Mountain Pass Theorem in order intervals, Leray–Schauder degree theory and the invariance of decreasing flow.

EXISTENCE OF MULTIPLE SOLUTIONS TO A FRACTIONAL DIFFERENCE BOUNDARY VALUE PROBLEM WITH PARAMETER

By establishing the corresponding variational framework, and using critical point theory, we give the existence of multiple solutions to a fractional difference boundary value problem with parameter. Under some suitable assumptions we obtain some results which ensure the existence of well precise interval of parameter for which the problem admits multiple solutions.

POSITIVE SOLUTIONS OF HIGHER DIMENSIONAL DISCRETE BOUNDARY VALUE PROBLEM

By means of the critical point theory, we prove the existence of positive solutions for a higher dimensional discrete boundary value problem. Our results generalize one of Agarwal in [2].

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS TO A SECOND-ORDER NONLINEAR DIFFERENCE SYSTEM

In this paper, by the critical point theory, a sufficient condition for the existence of multiple periodic solutions to a nonlinear second order difference system is obtained. An illustrative example is given. Our results improve and generalize some known ones.

Standing Waves for Discrete Nonlinear Schrodinger Equations with Nonperiodic Bounded Potentials

In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.