We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous...We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous real functions on R^N, and F ： R^N×R^2 → R is of class C^1. We assume that either F（x,ω） is super-quadratic and subcritical in ω∈R^2 or it is of the form ～1/P（x）｜ω｜^p ＋1/2^＊K（x）｜ω｜^2^＊ with p E （2,2^＊） and 2^＊ = 2N/（N-2）, the Sobolev critical exponent, P（x） and K（x） are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution ωε provided ε≤ε; and for any m∈N, there are m pairs of solutions ωε provided that ε≤εm and that F（x, ω） is,in addition, even in ω. Here ε and ωε are sufficiently small positive numbers. Moreover, the energy of ωε tends to 0 as ε→0.展开更多
We are concerned with a deformation theory in locally convex topological linear spaces. A special 'nice' partition of unity is given. This enables us to construct certain vector fields which are locally Lipsch...We are concerned with a deformation theory in locally convex topological linear spaces. A special 'nice' partition of unity is given. This enables us to construct certain vector fields which are locally Lipschitz continuous with respect to the locally convex topology. The existence, uniqueness and continuous dependence of flows associated to the vector fields are established. Deformations related to strongly indefinite functionals are then obtained. Finally, as applications, we prove some abstract critical point theorems.展开更多
In this paper we consider the first order discrete Hamiltonian systems {x1（n＋1）-x1（n）=Hx2（n,x（n））,x2（n）-x2（n-1）=Hx1（n,x（n））,where x（n） = （x2（n）x1（n））∑ R^2N, H（n,z） = 1/2S（n）z. z ＋ R（n,z...In this paper we consider the first order discrete Hamiltonian systems {x1（n＋1）-x1（n）=Hx2（n,x（n））,x2（n）-x2（n-1）=Hx1（n,x（n））,where x（n） = （x2（n）x1（n））∑ R^2N, H（n,z） = 1/2S（n）z. z ＋ R（n,z） is periodic in n and superlinear as {z} →4 ∞. We prove the existence and infinitely many （geometrically distinct） homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.展开更多
This paper deals via variational methods with the existence of infinitely many homoclinic orbits for a class of first order time dependent Hamiltonian systems=JH z(t,z)without any periodicity assumption on H, pro...This paper deals via variational methods with the existence of infinitely many homoclinic orbits for a class of first order time dependent Hamiltonian systems=JH z(t,z)without any periodicity assumption on H, providing that H(t,z) iseven with respect to z∈R 2N , superquadratic or subquadratic as |z|→∞, and satisfies some additional assumptions.展开更多
文摘We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous real functions on R^N, and F ： R^N×R^2 → R is of class C^1. We assume that either F（x,ω） is super-quadratic and subcritical in ω∈R^2 or it is of the form ～1/P（x）｜ω｜^p ＋1/2^＊K（x）｜ω｜^2^＊ with p E （2,2^＊） and 2^＊ = 2N/（N-2）, the Sobolev critical exponent, P（x） and K（x） are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution ωε provided ε≤ε; and for any m∈N, there are m pairs of solutions ωε provided that ε≤εm and that F（x, ω） is,in addition, even in ω. Here ε and ωε are sufficiently small positive numbers. Moreover, the energy of ωε tends to 0 as ε→0.
基金This work was supported by the National Natural Science Foundation of China(Grant No.19971091)the 973 Project of China.The author thanks the referees for their useful suggestions.
文摘We are concerned with a deformation theory in locally convex topological linear spaces. A special 'nice' partition of unity is given. This enables us to construct certain vector fields which are locally Lipschitz continuous with respect to the locally convex topology. The existence, uniqueness and continuous dependence of flows associated to the vector fields are established. Deformations related to strongly indefinite functionals are then obtained. Finally, as applications, we prove some abstract critical point theorems.
基金CHEN WenXiong supported by Science Foundation of Huaqiao UniversityYANG Minbo was supported by Natural Science Foundation of Zhejiang Province (Grant No. Y7080008)+1 种基金YANG Minbo was supported by National Natural Science Foundation of China (Grant No. 11101374, 10971194)DING Yanheng was supported partially by National Natural Science Foundation of China (Grant No. 10831005)
文摘In this paper we consider the first order discrete Hamiltonian systems {x1（n＋1）-x1（n）=Hx2（n,x（n））,x2（n）-x2（n-1）=Hx1（n,x（n））,where x（n） = （x2（n）x1（n））∑ R^2N, H（n,z） = 1/2S（n）z. z ＋ R（n,z） is periodic in n and superlinear as {z} →4 ∞. We prove the existence and infinitely many （geometrically distinct） homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.
文摘This paper deals via variational methods with the existence of infinitely many homoclinic orbits for a class of first order time dependent Hamiltonian systems=JH z(t,z)without any periodicity assumption on H, providing that H(t,z) iseven with respect to z∈R 2N , superquadratic or subquadratic as |z|→∞, and satisfies some additional assumptions.
