Infinitely many homoclinic solutions for some second order Hamiltonian systems

In this paper, we investigate the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems. By using fountain theorem due to Zou, we obtain two new criteria for guaranteeing that second order Hamiltonian systems have infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.     

Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.     

Infinitely many solutions for Hamiltonian systems

We consider two classes of the second-order Hamiltonian systems with symmetry. If the systems are asymptotically linear with resonance, we obtain infinitely many small-energy solutions by minimax technique. If the systems possess sign-changing potential, we also establish an existence theorem of infinitely many solutions by Morse theory.     

一类二阶Hamiltonian系统的无穷多周期解

研究一类超线性二阶Hamiltonian系统,且非线性项是奇的,不需要假设Ambros-etti-Rabinowitz的超二次条件,利用对称型山路引理得到无穷多周期解存在性结果.     

一类二阶Hamilton系统的周期解

本文研究一类含非定线性项的二阶Ｈａｍｉｌｔｏｎ系统多周期解问题．在位势函数满足超二次齐次条件下，利用临界点理论中对称型越山定理，证明了系统存在无穷多个给定周期的周期解．     

Infinitely many non-constant periodic solutions with negative fixed energy for Hamiltonian systems

ABSTRACT

In this paper, we obtain the existence of infinitely many non-constant periodic solutions with negative fixed energy for a class of second-order Hamiltonian systems, which greatly improves the existing results such as Zhang [Periodic solutions for some second order Hamiltonian systems. Nonlinearity. 2009;22(9):2141–2150, Theorem 1.5]. Moreover, we exhibit two simple and instructive examples to make our result more clear, which have not been solved by known results.     

Periodic Solutions of Classical Hamiltonian Systems without Palais-Smale Condition

In this paper, we study the existence of periodic solutions for classical Hamiltonian systems without the Palais-Smale condition. We prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions. Moreover, we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function V.     

Multiple periodic solutions for Hamiltonian systems with not coercive potential

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.     

Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in of the form , where we assume the existence of a sequence such that and as for any . Moreover, under a suitable non degeneracy condition, we prove that this class of systems admits multibump solutions. Received February 2, 1996 / In revised form July 5, 1996 / Accepted October 10, 1996     

A KAM Theorem for Reversible Systems of Infinite Dimension

For reversible systems of infinite dimension we prove an infinitely dimensional KAM theorem with an application to the network of weakly coupled oscillators of friction. The KAM theorem shows that there are many invariant tori of infinite dimension, and thus many almost periodic solutions, for the reversible systems.     

The existence of quasiperiodic solutions for coupled Duffing-type equations

In this paper, we prove the existence of infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations via KAM theorem. Moreover, the set of quasiperiodic solutions is of infinitely Lebesgue measure in the phase space.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions

The existence of infinitely many subharmonic solutions is obtained for a class of nonautonomous second order Hamiltonian systems with a new superquadratic condition. Furthermore, we can get the existence of homoclinic solutions as the limit of subharmonics under a stronger superquadratic condition which is still weaker than the growth conditions in the references.     

一类非线性薛定谔方程组无穷多解的存在性

讨论了一类非线性薛定谔方程组无穷多解的存在性.在假设的V(x),b(x),f(x)条件下,通过减弱喷泉定理的条件,运用变形的喷泉定理,证明了相关方程组的无穷多解的存在性.较扰动方法更加简捷.     

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF p（x）-BIHARMONIC EQUATIONS

In this paper,we study a class of p(x)-biharmonic equations with Navier boundary condition.Using the mountain pass theorem,fountain theorem,local linking theorem and symmetric mountain pass theorem,we establish the existence of at least one solution and infinitely many solutions of this problem,respectively.     

Elliptic system with some singular nonlinearity

We investigate multiplicity of solutions for elliptic systems with singular non-linearities. We obtain a theorem which shows that elliptic systems with some singular non-linearities have infinitely many solutions. We get this result by using the variational method, critical point theory and homology theory.     

Multiple solutions of nonlinear elliptic equations for oscillation problems

We discussed oscillating equations with Neumann boundary value in [Nonlinear Anal. 54 (2003) 431-443] and [J. Math. Anal. Appl. 298 (2004) 14-32] and prove the existence of infinitely many nonconstant solutions. However, it seems difficult to find infinitely many disjoint order intervals for oscillating equations with Dirichlet boundary value. To get rid of this difficulty, in this paper, we build up a mountain pass theorem in half-order intervals and use it to study oscillating problems with Dirichlet boundary value in which we only have the existence of super-solutions (or sub-solutions) and obtain new results on the exactly infinitely many solutions.     

THE SINGULAR SECOND ORDER NONLINEAR EIGENVALUE PROBLEM WITH INFINITELY MANY POSITIVE SOLUTIONS

1 IntroductionIn this paper we consider positive solutions for second order non1ineareigenVaue problem of the fOrmwhere A > 0, a,P,7,b 2 0 and p = 7P a7 6cr > 0.Positive solutions of nonlinear eigenvalue problem such as (P) have beenstudied extensiveIy by many authors (see tll-[61 and the references therein).However, to my best knowledge, less attention has been given to infinitelymany positive solutions. It must be pointed out that in all existing literaturesconcerning multiple positiv…