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We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$ : (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align} where a > 0, $b\geq0$ , and λ > 0 are constants, $\partial\Omega\neq\emptyset$ , $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$ , and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*} has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.

In this paper, by using a concentration-compactness argument, we study the existence of ground state solutions for nonlinear Schrodinger equations with asymptotically periodic potentials. In particular, when the coefficients are "competing", some sufficient conditions are given to guarantee the existence of ground state solutions, which improve and generalize some previous results in the literature.

We study a class of perturbed Schrodinger lattice systems without even conditions. Based on variational methods, we obtain the existence of two nontrivial homoclinic solutions. In particular, one of them is a ground state homoclinic solution, i.e., nontrivial homoclinic solution with least possible energy of this equation, and it is also a local minimizer of the action functional. To the best of our knowledge, there is no published result focusing on the systems.

In this paper, a nontrivial solution u ∈ H 1 ( R N ) to the autonomous Choquard equation with a local term − Δ u + λ u = ( I α ∗ | u | p ) | u | p − 2 u + | u | q − 2 u in R N is obtained, where N ≥ 3 , α ∈ ( 0 , N ) , λ > 0 is a constant, I α is the Riesz potential, N + α N p N + α N − 2 and q ∈ ( 2 , 2 ∗ = 2 N N − 2 ) . Under some further assumptions on p and q , the regularity and the Pohožaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation − Δ u + V ( x ) u = ( I α ∗ | u | p ) | u | p − 2 u + | u | q − 2 u in R N is also found under some assumptions on V ( x ) .

In this paper, we consider the following nonhomogeneous Schrodinger-Poisson equation $ (*)\left\{ \begin{align} & -\Delta u+V(x)u+\phi (x)u=-k(x)|u{{|}^{q-2}}u+h(x)|u{{|}^{p-2}}u+g(x),\ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ & -\Delta \phi ={{u}^{2}},\quad {{\lim }_{\text{ }\!\!|\!\!\text{ }x|\to +\infty }}\,\phi (x)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ \end{align} \right. $ where $ 1< q < 2, 4 < p < 6$. Under some suitable assumptions on $V(x), k(x), h(x)$ and $g(x)$, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

In this paper, we confirm the existence of three positive solutions for a class of Riemann-Liouville fractional $q$-difference equation which satisfies the boundary conditions. We gain several sufficient conditions for the existence of three positive solutions of this boundary value problem by applying the Leggett-Williams fixed point theorem.

We study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.

In this paper, we consider the following autonomous Fermi–Pasta–Ulam lattice dynamical system: $$\begin{aligned} \ddot{q}_{i}=\Phi '_{i-1}(q_{i-1}-q_{i})-\Phi '_{i}(q_{i}-q_{i+1}),\qquad i\in \mathbb {Z}, \end{aligned}$$ where $$\Phi _{i}$$ denotes the interaction potential between two neighboring particles and $$q_{i}(t)$$ is the state of the i-th particle. Supposing $$\Phi _{i}(x)$$ is asymptotically quadratic at infinity, i.e., $$\Phi _{i}(x)$$ tends to a quadratic function as $$|x|\rightarrow \infty $$ , for all $$T>0$$ , we obtain a nonzero T-periodic solution of finite energy. To our knowledge, there is no result dealing with this asymptotically quadratic case.

In this paper, we study a system of Schrodinger-Poisson equation {-∆u + a(x)u + K(x)ϕu=|u|p-2u,x ∈ R3,-∆ϕ=K(x)u2,x ∈ R3,} where p ∈ (4,6) and K ≥ (#8802;)0. Under some suitable decay assumptions but without any symmetry property on a and K, we obtain infinitely many solutions of this system.

In the whole space R N , we will study the following nonlinear elliptic system in two cases: − △ u + V 1 ( x ) u = f ( x , u , v ) , x ∈ R N , − △ v + V 2 ( x ) v = g ( x , u , v ) , x ∈ R N , u ( x ) → 0 , v ( x ) → 0 , | x | → ∞ . Case 1 : The periodic case (i.e., V 1 , V 2 , f and g are periodic in x i for i = 1 , … , N ), we obtain the nonexistence of nontrivial solutions for the system. The existence result has been obtained in Chen and Ma (2013). Case 2 : The non-periodic case (i.e., V 1 , V 2 , f and g are non-periodic), we mainly focus on the case where the nonlinearities f and g are superlinear at infinity, and we obtain infinitely many nontrivial solutions of the system by variational methods. To the best of our knowledge, there is no work focusing on the system in Cases 1–2. The main novelties of this paper can be summarized as follows: (1) The system is defined in the whole space R N ; (2) The potentials V i ( x ) ( i = 1 , 2 ) can be sign-changing; (3) The nonexistence of nontrivial solutions in the periodic case is obtained; (4) Infinitely many nontrivial solutions in the non-periodic case are obtained.

In this article, we study the following nonhomogeneous Schrodinger-Poisson equations { - Δ u + λ V ( x ) u + K ( x ) ϕ u = f ( x , u ) + g ( x ) , x ∈ R 3 , - Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where λ > 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be sign-changing.

In this paper, we consider the periodic discrete nonlinear equation { L u n − ω u n = ± g n ( u n ) , n ∈ Z , lim | n | → ∞ ⁡ u n = 0 , where L is a Jacobi operator, and the nonlinearities g n ( s ) are asymptotically linear as | s | → ∞ . In the two different cases (ω is a spectral endpoint of L, or it belongs to a finite spectral gap of L), we obtain the existence of nontrivial solitons of this equation by using variational methods. In particular, a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation. Here, solitons appear when we look for standing waves of some discrete nonlinear Schrodinger equations.

In the present paper, we consider the following Kirchhoff type problems { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = u 5 , in R 3 , u ∈ D 1 , 2 ( R 3 ) , where a > 0 , b > 0 and V ∈ L 3 2 ( R 3 ) is a given nonnegative function. If | V | 3 2 is suitable small, we prove that this problem has at least one bound state solution.

In the present paper, we consider the following Kirchhoff problem ( K ) { − ( 1 + b ∫ R 3 | ∇ u | 2 d x ) Δ u + a ( x ) u = | u | p − 2 u , in R 3 , u ∈ H 1 ( R 3 ) , where b > 0 and p ∈ ( 4 , 6 ) . Under some suitable decay assumptions but without any symmetry property on a ( x ) , we obtain infinitely many bound state solutions of this problem.

In this paper, we are concerned with the following nonlinear Schrodinger equations with hardy potential and critical Sobolev exponent \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), （1） \end{array} \right. \end{equation} where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu

In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equations q ¨ n = U ' ( q n + 1 − q n ) − U ' ( q n − q n − 1 ) , n ∈ ℤ , where q n denotes the displacement of the n -th lattice site and U is the potential of interaction between two adjacent particles. Inspired by previous work due to Szulkin and Weth (Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822), we investigate the existence of solitary ground waves, i.e., nontrivial solutions with least possible energy.

In this paper, we study the invasion dynamics of a diffusive pioneer-climax model in monotone and non-monotone cases. For parameter ranges in which the system admits monotone properties, we establish the existence of spreading speeds and their coincidence with the minimum wave speeds by monotone dynamical system theories. The linear determinacy of the minimum wave speeds is also studied by constructing suitable upper solutions. For parameter ranges in which the system is non-monotone, we further determine the existence of spreading speeds and traveling waves by the sandwich technique and upper-lower solution method. Our results generalize the existing results established under monotone assumptions to more general cases.

In this paper, an autonomous Choquard equation with doubly critical exponents is studied. By using the Pohozaev constraint and the perturbed method, a positive and radially symmetric ground state solution in $H^1(\mathbb {R}^N)$ is obtained. The result here extends and complements the earlier theorems obtained by Seok and Moroz and Schaftingen.

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,&{}\quad x\in \mathbb {R}^N,\\ \\ u\in H^1(\mathbb {R}^N),\quad u>0,&{}\quad x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where $$N\ge 3$$ , $$\max \{0,N-4\}0,b\ge 0$$ are constants, $$I_{\alpha }$$ is the Riesz potential and $$V{:}\,\mathbb {R}^N\rightarrow \mathbb {R}$$ is a potential function. Under certain assumptions on V, we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.

In this paper, we consider the following Choquard equation $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \Delta u+a(x)u=(I_\alpha *|u|^p)|u|^{p-2}u, &{}x\in {\mathbb {R}}^N,\\ u(x)\rightarrow 0,&{}|x|\rightarrow +\infty , \end{array}\right. {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\hbox {CH})} \end{aligned}$$ where $$N\ge 3, I_\alpha $$ is a Riesz potential, $$\frac{N+\alpha }{N}

In this article, we give a new proof on the existence of infinitely many sign-changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential − Δ u − μ u | x | 2 = λ u + | u | 2 * − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a smooth open bounded domain of which contains the origin, 2 * = 2 N N − 2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7 , μ ∈ [ 0 , μ ¯ − 4 ) , and μ ¯ = ( N − 2 ) 2 4 , we show that the problem admits infinitely many sign-changing solutions for each fixed λ > 0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.

In this paper, we consider the following nonlinear Schrodinger–Poisson system { − Δ u + V ( x ) u + ϕ u = f ( x , u ) , in R 3 , − Δ ϕ = u 2 , in R 3 , where the nonlinearity f is superlinear at infinity with subcritical or critical growth and V is positive, continuous and periodic in x . The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this system is obtained. Moreover, when V ≡ 1 , we obtain ground state solutions for the above system with a wide class of superlinear nonlinearities by using a new approach.

The existence and concentration of positive solutions for a class of Kirchhoff-type problems with a nonnegative steep well potential in R 3 are obtained via variational methods.

We consider the existence of periodic and unbounded solutions for the asymmetric oscillator x ″ + a x + − b x − + g ( x ) = p ( t ) , where x + = max ⁡ { x , 0 } , x − = max ⁡ { − x , 0 } , a and b are two positive constants, p ( t ) is a 2π-periodic smooth function and g ( x ) satisfies lim | x | → + ∞ ⁡ x − 1 g ( x ) = 0 . We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φ p ( θ ) and Λ ( h ) . In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π-periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman–Lazer's and Fabry–Fonda's existence conditions. In contrast with many existing results, the function g ( x ) may be unbounded or oscillatory without asymptotic limits.

In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\lambda|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq 3,\ \alpha\in(0,N)$, $\lambda>0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+\alpha}{N}$ or $\frac{N+\alpha}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_\alpha$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Poho\v{z}aev constraint method, we obtain a positive and radially nonincreasing groundstate solution in $H^1(\mathbb{R}^N)$ for the problem. To the end, the regularity and the Poho\v{z}aev identity of solutions to a general Choquard equation are obtained.

In this paper, we study a system of Schrodinger–Poisson equation $$\left\{\begin{array}{ll}-\Delta u+(\lambda a(x)+a_0(x))u+K(x)\phi u=|u|^{p-2}u, & \quad x \in\mathbb{R}^3, \\- \Delta \phi=K(x)u^2,& \quad x \in \mathbb{R}^3 \end{array} \right.$$ where $${p \in (4,6)}$$ and $${\lambda}$$ is a parameter. We require that $${a(x) \geq 0}$$ and has a bounded potential well $${\Omega = a^{-1}(0)}$$ . Combining this with other suitable assumptions on Ω, a 0 and K, we obtain the existence of multi-bump-type solution $${u_\lambda}$$ when $${\lambda}$$ is large via variational methods.

In this paper, we are concerned with elliptic equations with Hardy potential and critical Sobolev exponents where $2^{*}={2N}/({N-2})$ is the critical Sobolev exponent, $N\geq 3$, $0\leq \mu \lt \overline {\mu }={(N-2)^2}/{4}$, $\mathbf {\Omega }\subset \mathbb {R}^{N}$ an open bounded set. For $\lambda \in [0,\lambda _{1})$ with $\lambda _{1}$ being the first eigenvalue of the operator $-\Delta -{\mu }/{|x|^{2}}$ with zero Dirichlet boundary condition, and for $f\in H_{0}^{1}(\mathbf {\Omega })^{-1}=H^{-1}$, $f\neq 0$, we show that (\ref {eq1}) admits at least two distinct nontrivial solutions $u_{0}$ and $u_{1}$ in $H_{0}^{1}(\mathbf {\Omega })$. Furthermore, $u_{0}\geq 0$ and $u_{1}\geq 0$ whenever $f\geq 0$.

In this paper, we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q ¨ i ( t ) = Φ i − 1 ′ ( t , q i − 1 ( t ) − q i ( t ) ) − Φ i ′ ( t , q i ( t ) − q i + 1 ( t ) ) , i ∈ Z , where Φ i ( t , x ) = − ( α i / 2 ) | x | 2 + V i ( t , x ) is T-periodic in t for T > 0 and satisfies Φ i + N = Φ i for some N ∈ N , q i ( t ) is the state of the i-th particle. Assume that α i = 0 for some i ∈ Z and V i ′ ( t , x ) denoting the derivative of V i respect to x is asymptotically linear with x both at origin and at infinity. We would like to point out that this system is resonant both at origin and at infinity and not studied up to now. Based on some new results concerning the precise computations of the critical groups, for a given m ∈ Z , we obtain the existence of nontrivial periodic solutions satisfying q i + m N ( t + T ) = q i ( t ) for all t ∈ R and i ∈ Z under some additional conditions.

In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equations $$\begin{aligned} \ddot{q}_n= U^{\prime }(q_{n+1}-q_n)-U^{\prime }(q_n-q_{n-1}),\quad n\in \mathbb {Z}, \end{aligned}$$ where $$q_n$$ denotes the displacement of the $$n$$ -th lattice site and $$U$$ is the potential of interaction between two adjacent particles. We investigate the existence of two kinds travelling wave solutions: periodic and solitary ones under some growth conditions on $$U$$ which is different from the widely used Ambrosetti–Rabinowitz condition.

In this paper, we present a new proof on the existence of infinitely many sign-changing solutions for the following Brezis-Nirenberg problem \begin{eqnarray} -\Delta u=\lambda u+|u|^{2^{*}-2}u \quad \textrm{in}\, \Omega, \qquad u=0 \quad \textrm{on}\,\partial\Omega, \end{eqnarray} for each fixed $\lambda>0$, under the assumptions that $N\geq 7$, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent. In order to construct sign-changing solutions, we will use a combination of invariant sets method and Ljusternik-Schnirelman type minimax method, which is much simpler than the proof of [20] depending on the estimates of Morse indices of nodal solutions to obtain the same result.

In this paper, we consider the following Choquard equation ( CH ) − Δ u + u = ( I α ∗ | u | p ) | u | p − 2 u + V ( x ) | u | q − 2 u in R N , u ∈ H 1 ( R N ) , where N ≥ 3 , α ∈ ( ( N − 4 ) + , N ) , p ∈ [ 2 , N + α N − 2 ) , q ∈ ( 2 , 2 N N − 2 ) ∩ ( 1 + ( p − 1 ) ( N − 2 ) α N 2 , 1 + 2 N ( N − α ) + N 2 ( p − 1 ) ( N − 2 ) α ) and I α is the Riesz potential. Under some suitable decay assumptions but without any symmetry property on V ( x ) , we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large.

In this paper, we study a class of resonant cooperative elliptic systems with sublinear or superlinear terms and obtain infinitely many nontrivial solutions by two variant fountain theorems developed by Zou.

We consider the boundedness and unboundedness of solutions for the asymmetric oscillator $$\begin{aligned} x''+ax^+-bx^-+g(x)=p(t), \end{aligned}$$ where $$x^+=\max \{x,0\},x^-=\max \{-x,0\}, a$$ and $$b$$ are two positive constants, $$ p(t)$$ is a $$2\pi $$ -periodic smooth function and $$g(x)$$ satisfies $$\lim _{|x|\rightarrow +\infty }x^{-1}g(x)=0$$ . We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. It turns out that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions $$\Phi _p(\theta )$$ and $$\Lambda (h)$$ . Some explicit conditions are given for the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function $$g(x)$$ is required to be a bounded function with asymptotic limits, here we allow $$g(x)$$ be unbounded or oscillatory without asymptotic limits.

In this paper, we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equations q ¨ i = Φ i − 1 ′ ( q i − 1 − q i ) − Φ i ′ ( q i − q i + 1 ) , i ∈ Z , where Φ i denotes the interaction potential between two neighboring particles and q i ( t ) is the state of the i -th particle. Supposing Φ i is superquadratic at infinity, for all T > 0 , we obtain a nonzero T -periodic solution of finite energy which may be nonconstant in some range of period. If in addition Φ i ( x ) is even in x , we also obtain infinitely many geometrically distinct solutions for any period T > 0 . In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group Z and the continuous compact group S 1 under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by Bartsch and Ding [Math. Nachr. 279 (2006) 1267–1288] to a more general class of symmetry.

In this paper, the existence of nontrivial solutions is obtained for a class of Kirchhoff type problems with Dirichlet boundary conditions by computing the critical groups and Morse theory.

We study the discrete nonlinear equation where (the spectrum of L) and is asymptotically linear as for all . We obtain the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons of this equation. Our method is based on the generalized Nehari manifold method developed recently by Szulkin and Weth. To the best of our knowledge, this technique has not been used for discrete equations with saturable nonlinearities.

In this paper, we consider the second order Hamiltonian system $$\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= −d2/dt2 −A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.

In this paper, we study the existence of positive ground state solutions for non-autonomous Kirchhoff type problems: $$ -\Big (1+b\int _{\mathbb R^3} |\nabla u|^2\Big ) \Delta u+u=a(x)|u|^{p-1}u \quad \mbox {in } \mathbb {R}^3, $$ where $b>0$, $3\lt p\lt 5$ and $a:\mathbb R^3\rightarrow \mathbb R$ is such that $$ \lim _{|x|\rightarrow \infty } a(x)=a_\infty >0, $$ but no symmetry property on $a(x)$ is required.

Under the assumption that F is asymptotically or super linear as | U |→∞ with U =( u, v ) ∈ R 2 , we obtain the existence of ground state solutions of a class of cooperative elliptic systems in R N by using a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. To the best of our knowledge, there is no result published concerning the systems in the whole space R N .

In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.

In this paper, we study the existence of multibump solutions for discrete nonlinear Schrodinger equations with periodic potentials. We first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group. Then, we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear Schrodinger equations, and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case. As an application, we get, without any symmetry assumptions, infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity.

In this paper, we consider the superquadratic second order Hamiltonian system u ″ ( t ) + A ( t ) u ( t ) + ∇ H ( t , u ( t ) ) = 0 , t ∈ R . Our main results here allow the classical Ambrosetti–Rabinowitz superlinear condition to be replaced by a general superquadratic condition, and 0 lies in a gap of σ ( B ) , where B : = − d 2 d t 2 − A ( t ) . We will study the ground state periodic solutions for this problem. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou.