Junxiang Xu

Southeast University · Department of Mathematics

In this paper, we consider a one-dimensional completely degenerate oscillator subjected to an analytically \(\epsilon \)-dependent quasi-periodic perturbation, whose frequencies satisfy a Diophantine condition. By the KAM method, we show that one of the following results holds true: 1. For all sufficiently small \(\epsilon \) and all initial values...

This paper concerns a quasi-periodically forced elliptic-type degenerate harmonic oscillator equation: x¨+x2d+1=f(ϵ,ωt,x,x˙),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docume...

This paper considers a class of 3-dimensional real analytic nonlinear quasi-periodic systems with a small perturbation parameter, whose unperturbed part has a degenerate equilibrium point. Based on the Leray-Schauder Continuation Theorem [18], [20] and using technique of outer parameter, we prove that the system has a small response solution for ma...

In this paper, two dimensional modified Boussinesq equation $$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions correspon...

We deal with the singularly perturbed Choquard equation in the plane with critical exponential growth in the sense of the Trudinger–Moser inequality. Under a local condition imposed on the potential, we show the multiplicity and concentration of positive solutions via penalization technique and variational methods.

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-deg...

In this paper, we study the following semilinear Kirchhoff type equation: −ϵ2a+ϵb∫R3|∇u|2Δu+V(x)u=f(u)+u5,u>0, u∈H1(R3), where ϵ>0 is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attain...

We are concerned with the Schrödinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{lll} -\,\epsilon ^2\Delta u+V(x)u+K(x)\phi u=u^5,&{}\quad x\in {\mathbb {R}}^3,\\ -\,\Delta \phi =K(x)u^2, &{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$where \(\epsilon >0\) is a parameter, \(V\in L^{\frac{3}{2}}({\mathbb {R}}^3)\) and \...

This work focuses on higher-dimensional quasi-periodically forced nonlinear beam equation. This means studying $$\begin{aligned} u_{tt} + ( -\Delta +M_\xi )^2u +\varepsilon \phi (t) ( u+u^3 ) =0, \quad x\in \mathbf {R}^d, t\in \mathbf {R} \end{aligned}$$with periodic boundary conditions, where \(\varepsilon \) is a small positive parameter, \(\phi...

In this paper we consider the system x = (A() + mP(t; ))x, x ∈ R³, where is a small parameter, A, P are all 3 × 3 skew symmetric matrices, A is a constant matrix with eigenvalues ±iλ¯() and 0, where λ¯() = λ + am0m⁰ + O(m0⁺¹)(m0 < m), am0 = 0, P is a quasi-periodic matrix with basic frequencies ω = (1, α) with α being irrational. First, it is prove...

In this paper we prove the persistence of invariant tori of analytic reversible systems under Brjuno–Rüssmann’s non-resonant condition by an improved KAM iteration, but the frequency may undergo some drift. Furthermore, if the frequency mapping has nonzero Brouwer’s topological degree, then the invariant torus with prescribed frequency can persist....

In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Mo...

In this paper, one-dimensional quasi-periodically forced generalized boussinesq equation (Iquestion Presented) with hinged boundary conditions is considered, where ϵ is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω1, ω2, ··· ωm). It is proved that, under a suitable hypothesis on φ(t),...

In this paper, we study a generalized Choquard equation −Δu+V(x)u=(∫RNQ(y)F(u(y))|x−y|μdy)Q(x)f(u),u∈H1(RN), where 0<μ<N, V and Q are linear and nonlinear potentials, and F is the primitive function of f. When the potentials are periodic and f is odd or even, we find infinitely many geometrically distinct solutions using the method of Nehari manifo...

For a class of generalized Choquard equations with perturbation terms, we establish the existence of bound and ground states by virtue of the method of Nehari manifold.

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, and prove that if the frequency mapping Omega;(y) ? ?? and normal frequency mapping ?(y) ? ? satisfy that deg (w/?, O,w0/?0 ? 0, where ?0 = ?(y0) and ?0 = ?(y0) satisfy Melnikov's non-resonance conditions for some...

In this paper we are mainly concerned with the persistence of invariant tori with prescribed frequency for analytic nearly integrable Hamiltonian systems under the Brjuno–Rüssmann non-resonant condition, when the Kolmogorov non-degeneracy condition is violated. As it is well known, the frequency of the persisting invariant tori may undergo some dri...

This paper puts forward a new KAM theorem for the hyperbolic lower dimensional tori in reversible system without assuming any non-degeneracy condition, which can be viewed as a unified form under all kinds of non-degenerate conditions, including Kolmogorov, Bruno and Rüssmann non-degeneracy condition. The theorem with some non-degeneracy conditions...

For a class of asymptotically periodic quasilinear Schrödinger equations with three times growth, the existence and nonexistence of ground states are established. The method used here is based on the method of Nehari manifold and concentration compactness principle. Copyright

In this paper, we are concerned with superlinear fractional Schrödinger equation (−Δ) s u + V(x) u = f(x, u), x ∈ ℝ N , where f is continuous. When V and f are asymptotically periodic in x, we show the existence of ground states. When V and f are periodic in x, we obtain infinitely many geometrically distinct solutions. The method used here is bas...

In this paper, we consider the effective reducibility of the following quasi-periodic Hamiltonian system $$\begin{aligned} \dot{x}=(A+\varepsilon Q(t,\varepsilon ))x,~|\varepsilon |\le \varepsilon _0, \end{aligned}$$where A is a constant matrix with different eigenvalues, \(Q(t,\varepsilon )\) is analytic quasi-periodic on \(D_\rho \) with respect...

This work focuses on the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. By KAM method and the special structure of unperturbed nonlinear terms, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of [22].

In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2} \Delta u+a(x)u+\lambda\phi(x)u=b(x)f(u)+|u|^{4}u,&x\in\mathbb{R}^{3}, \\ -\varepsilon^{2} \Delta\phi=u^{2},\ u\in H^{1}(\mathbb{R}^{3}),&x\in\m...

In this paper we prove the persistence of invariant tori for analytic perturbation of constant vector field under weaker non-degeneracy condition. In the proof we introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function \({q^{n} \epsilon}\), \({0 , rather than super exponential function.

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we ca...

In this paper, we study the existence, multiplicity, and concentration of positive solutions for the semilinear Schrödinger equation − ε 2 Δ u + K ( x ) u = Q ( x ) u p − 2 u + f ( u ) , u ∈ H 1 ( R N ) , where ε > 0 is a small parameter, N ≥ 3 and 2 < p < 2 ∗ = 2 N N − 2 , K and Q are positive continuous functions, f is a continuous superlinear no...

This paper considers a class of nearly integrable twist symplectic mappings with generating functions and proves the persistence of lower dimensional elliptic invariant tori under Rüssmann’s non-degeneracy conditions.

In this paper, one-dimensional generalized Boussinesq equation: utt − uxx + (u 2 + uxx ) xx = 0 with boundary conditions ux (0, t) = ux (π, t) = uxxx (0, t) = uxxx (π, t) = 0 is considered. It is proved that the equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with 2-dimensional Diophantine frequencies. The proof...

In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed hamiltonian system, $$H=\langle \omega(\xi), I \rangle +\frac12 \Omega_0(u^2+v^2)+P(\theta,I,z,\bar{z};\xi)$$ and prove that if frequencies $(\omega_0,\Omega_0)$ satisfy some non-resonant conditions and the Brouwer degree of the frequency mapping $\...

Using the method of the Nehari manifold, we study the existence of ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part and superlinear nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, one-dimensional derivative Schrödinger equation, with periodic boundary condition is considered. It is proved that the above equation admits a Whitney smooth family of small amplitude, quasi-periodic solutions with two-dimensional Diophantine frequencies. The proof is based on infinite-dimensional Kolmogorov-Arnold-Moser (KAM) theory...

In this paper, one-dimensional defocusing modified KDV equation: u(t) + u(xxx) - 6u(2)u(x) = 0 with periodic boundary condition is considered. It is proved that the above equation admits a Cantor family of small amplitude, quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and partial Birkhoff normal form.

In this paper we prove the existence of invariant curves for analytic reversible mappings under Brjuno–Rüssmann’s non-resonant condition. In the proof we use the polynomial structure of function to truncate, introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function (Formula Presented.) rather than super e...

In this article we study the existence and nonexistence of ground states of the Schrodinger-Poisson system $$\displaylines{ -\Delta u+V(x)u+K(x)\phi u=Q(x)u^3,\quad x\in \mathbb{R}^3,\cr -\Delta\phi=K(x)u^2, \quad x\in \mathbb{R}^3, }$$ where V, K, and Q are asymptotically periodic in the variable x. The proof is based on the the method of Nehari m...

In this paper, we consider Hamilton–Jacobi equations with homogeneous Neumann boundary condition. We establish some results on noncompact manifold with homogeneous Neumann boundary conditions in view of weak Kolmogorov-Arnold-Moser (KAM) theory, which is a generalization of the results obtained by Fathi under the non-bounded condition. Copyright ©...

In this paper we consider a class of nearly integrable symplectic mappings with generating function and prove the persistence of lower dimensional hyperbolic invariant tori. Under a Rüssmann-type non-degenerate condition, by using the KAM theory, we proved that the nearly integrable twist symplectic mappings admit a family of lower dimensional inva...

This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small pertur...

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the s...

In this paper a one-dimensional generalized Boussinesq equation utt−uxx+(u3+uxx)xx=0utt−uxx+(u3+uxx)xx=0

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous dif...

In this paper, we consider real analytic nearly integrable hamiltonian systems under Bruno non-degeneracy condition, and prove that if the rank of the Jacobian of the n-dimensional frequency vectors with respect to action variables is n-1, then there exists a one-parameter analytic family of invariant tori, whose frequencies are the small dilation...

For a class of asymptotically periodic Schrodinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.

In this paper, we are concerned with the existence of ground state solutions for Schrödinger equation −Δu+V(x)u=K(x)|u|2⁎−2u+f(x,u),u∈H1(RN), N⩾4N⩾4 V x −Δ+V−Δ+V K f x

In this paper we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system, and prove that if the frequency mapping has nonzero topological degree at some Diophantine frequency, then the hyperbolic lower invariant dimensional torus with this frequency persists under small perturbations.

This paper considers the reducibility and existence of periodic solutions for a class of nonlinear periodic system with a degenerate equilibrium point under small perturbations. By introducing some parameter, we consider an equivalent periodic system. Then we prove that by an affine linear periodic transformation the parameterized periodic system i...

In this paper, one-dimensional generalized Boussinesq equation with hinged boundary conditions is considered, where . It is proved that for each prescribed integer , the above equation admits a Whitney-smooth family of small amplitude, quasi-periodic solutions with -dimensional Diophantine frequencies. The proof is based on an infinite-dimensional...

We prove the existence of positive ground states for the nonlinear Schrödinger system { − Δ u + ( 1 + a ( x ) ) u = F u ( u , v ) + λ v , − Δ v + ( 1 + b ( x ) ) v = F v ( u , v ) + λ u , where a , b are periodic or asymptotically periodic and F satisfies some superlinear conditions in ( u , v ) . The proof is based on the method of Nehari manifold...

In this paper, we establish the local well-posedness for the two-component b-family system in a range of the Besov space. We also derive the blow-up scenario for strong solutions of the system. In addition, we determine the wave-breaking mechanism to the two-component Dullin–Gottwald–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.

Using the Nehari manifold and the concentration compactness principle, we study the existence of ground state solutions for asymptotically periodic Schrodinger equations with critical growth.

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{arra...

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrodinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{arr...

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state so...

In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a n...

In this paper, we study the existence of multi-bump solutions for the semilinear Schr?dinger?Poisson system where p???(1, 5), and a(x)?>?0, b(x)?>?0 in . For any positive integer K, we prove that there exists ?(K)?>?0 such that, for 0?<???<??(K), the system has a K-bump solution. Then the equation has more and more multi-bump solutions as ????0.

In this paper, we establish the existence of positive ground states for asymptotically periodic Schrödinger–Poisson systems with general nonlinearities by the method of Nehari manifold. Copyright © 2012 John Wiley & Sons, Ltd.

In this article we study the initial-value problem for the periodic two-component b-family system, including a special case, when b=2, which is referred to as the two-component Dullin-Gottwald-Holm (DGH) system. We first show that the two-component b-family system can be derived from the theory of shallow-water waves moving over a linear shear flow...

In this paper we concern with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation{−(ε2a+bε∫R3|∇u|2)Δu+M(x)u=λf(u)+|u|4u,x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0ε>0 is a small parameter, a, b are positive constants and λ>0λ>0 is a parameter, and f is a continuous superlinear and subcritical nonlinearity. Supp...

In this paper, we study the following nonperiodic Hamiltonian system ż=JHz(t,z), where H(t,z)H(t,z) is superquadratic in z∈R2Nz∈R2N as |z|→∞|z|→∞. By applying a generalized linking theorem for strongly indefinite functionals, we obtain infinitely many homoclinic orbits for the above system.

This paper is concerned with the following semilinear elliptic equations of the form where ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε ) for all sufficiently small ε > 0. Moreover, we show...

Considered herein is the well-posedness problem of the periodic two-component Dullin–Gottwald–Holm (DGH) system on the circle, which can be derived from Eulerʼs equation with constant vorticity in shallow water waves moving over a linear shear flow. The result of blow-up solutions for certain initial profiles in a manner which corresponds to wave-b...

In this paper we study the persistence properties of decay for the solutions to the two-component b-family system. Using the method of characteristics, we establish that certain decay properties of the initial data persist as long as the solution exists. We also examine the propagation behavior of compactly supported solutions. We show that solutio...

In this paper we study the following nonlinear Schrödinger equation where a is periodic with respect to x and f(x, u) is superlinear in u. Suppose that 0 lies in a gap of the spectrum σ( − Δ + a). Without periodic assumption on f, we prove the existence of ground state solution for the system. Moreover, we obtain some sufficient conditions for the...

In this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems. The normal variable is two-dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure.

In this paper we study the first order nonautonomous Hamiltonian system ż = JH z(t, z), where H(t, z) depends periodically on t. By using a generalized linking theorem for strongly indefinite functionals, we prove that the system has infinitely many homoclinic orbits for weak superlinear cases. © 2012 Juliusz Schauder University Centre for Nonlinea...

In this paper we consider the persistence of lower dimensional invariant tori for a class of reversible systems, and prove that if the average of the linear terms has full rank, then the invariant torus with Diophantine frequencies persists under small perturbations. KeywordsReversible systems–KAM iteration–Invariant tori

In this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

This paper is concerned with the following Hamiltonian elliptic system: {−△u+V(x)u=Hu(x,u,v)in RN;−△v+V(x)v=−Hv(x,u,v)in RN;u(x)→0andv(x)→0as |x|→∞, where V(x)∈C(RN,R) and H(x,z) is superquadratic in z as |z|→∞ with z:=(u,v). By applying a critical point theorem for strongly indefinite functionals, we obtain infinitely many solutions for the above...

Regrettably, the last two words of the article’s title were omitted in the original version [J. Wang, J. Xu, F. Zhang and L. Wang, NoDEA, Nonlinear Differ. Equ. Appl. 17, No. 4, 411–435 (2010; Zbl 1202.58015)]. The correct article title is “Homoclinic orbits for unbounded superquadratic Hamiltonian systems”.

We consider the following diffusion system: {∂tu−△xu+b(t,x)∇xu+V(x)u=Hv(t,x,u,v),−∂tv−△xv+b(t,x)∇xv+V(x)v=Hu(t,x,u,v)∀(t,x)∈R×RN, which is an unbounded Hamiltonian system in L2(R×RN,R2m), z:=(u,v):R×RN→Rm×Rm, b∈C(R×RN,RN), V∈C(RN,R) and H∈C1(R×RN×R2m,R). Suppose that H,b and V depend periodically on t and x, and that H(t,x,z) is superquadratic in z...

This paper is concerned with solutions to the Dirac equation: −iΣαk∂ku + aβu + M(x)u = g(x, ‖u‖)u. Here M(x) is a general potential and g(x, ‖u‖) is a self-coupling which grows super-quadratically in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equat...

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic...

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system $\dot{x}=J{\nabla }_{x}H$ , where $H\left(x,t,\epsilon \right)=\left(1/2\right)\beta \left({x}_{1}^{2}+{x}_{2}^{2}\right)+F\left(x,t,\epsilon \right)$ with $\beta \ne 0,{\partial }_{x}F\left(0,t,\epsilon \right)=O\left(\epsilon \right)$ and ${\partia...

In this paper we study the following nonperiodic second order Hamiltonian system -u(t) + L(t)u(t) = del R-u(t, u(t)), for all (t, u) is an element of R x R-N, where the matrix L(t) is an element of C(R, R-N2) and R(t, u) is asymptotically quadratic or super quadratic in u as vertical bar u vertical bar -> infinity. Under more general assumptions on...

We study the following second-order differential equation: ( Φ p ( x ' ) ) ' + F ( x , t ) x ' + ω p Φ p ( x ) + α | x | l x + e ( x , t ) = 0 , where Φ p ( s ) = | s | ( p - 2 ) s   ( p > 1 ), α > 0 and ω > 0 are positive constants, and l satisfies - 1 < l < p - 2 . Under some assumptions on the parities of F ( x , t ) and e ( x , t ) , by a small...

We consider the persistence of lower dimensional elliptic invariant tori with prescribed frequencies in reversible systems, and prove that if the frequency mapping has non-zero Brouwer’s degree at a certain point that satisfies Melnikov’s non-resonance conditions, then the invariant torus with given frequency persists under small perturbations.

We study the existence of even homoclinic orbits for the second-order Hamiltonian system ü+Vu(t, u)=0. Let V(t, u)=−K(t, u)+W(t, u)∈C1(ℝ × ℝn, ℝ), where K is less quadratic and W is super quadratic in u at infinity. Since the system we considered is neither autonomous nor periodic, the (PS) condition is difficult to check when we use the Mountain P...

We consider the following nonperiodic diffusion systems $$ \left\{\begin{array}{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \end{array}\right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N}, $$where \({b\in C(\mathbb{R}\times\mathbb{R}^{N},\mat...

This paper is concerned with the following nonperiodic Hamiltonian elliptic system {-epsilon(2) Delta u + V(x)u = H-u(x, u, u) in R-N, -epsilon(2) Delta v + V(x)v = -H-v(x, u, v) in R-N, u(x) -> 0 and v(x) -> 0 as vertical bar x vertical bar -> infinity, where epsilon > 0 is a small parameter, and the potential V is bounded below, and H is asymptot...

In this paper we consider analytic nearly integrable hamiltonian systems, and prove that if the frequency mapping has nonzero Brouwer topological degree at some Diophantine frequency, then the invariant torus with this frequency persists under small perturbations.

In this paper we study the following nonperiodic second order Hamiltonian system−u¨(t)+L(t)u(t)=∇uR(t,u),∀t∈R, where L(t) may not be uniformly positive definite for all t∈R, and R(t,u) is asymptotically quadratic in u as |u|→∞. By applying a generalized linking theorem in Bartsch and Ding (2006) [8], we obtain the existence and multiplicity of homo...

This paper is concerned with solutions to the Dirac equation: −i∑αk∂ku+aβu+M(x)u=Ru(x,u). Here M(x) is a general potential and R(x,u) is a self-coupling which is super-quadratic in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct l...

This paper is concerned with solutions of the Hamiltonian system: , where with L being a 2N×2N symmetric matrix and W∈C 1(ℝ×ℝ2N ,ℝ) being super quadratic at infinity in u∈ℝ2N . We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we constructed linking le...

We study the existence of homoclinic solutions for the second order Hamiltonian system u ¨+V u (t,u)=f(t). Let V(t,u)=-K(t,u)+W(t,u)∈C 1 (ℝ×ℝ n ,ℝ) be T-periodic in t, where K is a quadratic growth function and W may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence...

This paper is concerned with the following nonperiodic Hamiltonian elliptic system {−Δu+V(x)u=Rv(x,u,v)in RN,−Δv+V(x)v=Ru(x,u,v)in RN,u(x)→0andv(x)→0as |x|→∞, where R(x,z) is superquadratic in z as |z|→∞ with z=(u,v). By applying a critical point theorem for strongly indefinite functionals, we prove the existence of solution for the above system.

This paper considers the existence of periodic solutions for a class of non-autonomous differential delay equations (*) x′(t) = -Σi=1n-1 f(t, x(t - iτ)), where τ > 0 is a given constant. It is shown that under some conditions on f and by using symplectic transformations, Floquet theory and some results in critical point theory, the existence of sin...

In this paper we prove Gevrey smoothness of the persisting invariant tori for small perturbations of analytic linear reversible systems with Russmann's non-degeneracy condition by an improved KAM iteration method with parameters.

We establish the existence and multiplicity of solutions for the semiclassical nonlinear Schrödinger equation{−ε2Δu+V(x)u=g(x,u)for x∈RN,u(x)→0as |x|→∞, where ε>0 is a small parameter, and the potential V changes sign and may not be bounded from below, and g(x,u) is asymptotically linear in u as |u|→∞.

This paper is concerned with solutions to the Dirac equation: −iΣα k ∂ k u + aβu + M(x)u = g(x, ‖u‖)u. Here M(x) is a general potential and g(x, ‖u‖) is a self-coupling which grows super-quadratically in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dir...

We consider small perturbation of analytic reversible mappings with degeneracy and prove the existence of invariant curve by KAM iteration. Moreover, the frequency of invariant curve persists without any drift.

This paper concerns solutions for the Hamiltonian system: z˙=𝒥Hz(t,z). Here H(t,z)=(1/2)z⋅Lz+W(t,z), L is a 2N×2N symmetric matrix, and W∈C1(ℝ×ℝ2N,ℝ). We consider the case that 0∈σc(−(𝒥(d/dt)+L)) and W satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theor...

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system: ẋ=Ax+f(t,x,ϵ),x∈R2 where A is a real 2×2 constant matrix, and f(t,0,ϵ)=O(ϵ) and ∂xf(t,0,ϵ)=O(ϵ) as ϵ→0. With some non-resonant conditions of the frequencies with the eigenvalues of A and without any nondegeneracy condition with respect to ϵ, by an...

In this paper we study the boundedness of solutions for the second-order differential equation (Fp(x′)+ψ(x′))′+αFp(x+)−βFp(x−)=φ(t,x), where Fp(s)=|s|p−2s, p>1 and α, β are strictly positive constants satisfying a resonant relation with n being a positive integer, and φ(t,x) is a 2π-periodic function in t. There exists a function W(θ) such that if...

In this paper with the KAM iteration we prove a KAM theorem for nearly integrable Hamiltonian systems with two degrees of freedom without any non-degeneracy condition.

We consider Gevrey-smooth perturbation of a completely integrable Gevrey-smooth Hamiltonian under Rüssmann’s non-degeneracy condition, and obtain a family of invariant tori using an improved KAM iteration; they are Gevrey smooth in the sense of Whitney, with a Gevrey index depending on the Gevrey class of the Hamiltonian and on the exponent in the...

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann’s non-degeneracy condition, w...