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In this paper, we are concerned with the following X-ray free electron laser Schrödinger equation (i ∇ - A (x)) 2 u + V (x) u - μ | x | u = g (x , | u |) u , x ∈ R N , N ≥ 2 , where V is the electric potential, A is the magnetic potential and μ ≥ 0 is a constant. We mainly study the existence/nonexistence of ground states and their asymptotic behavior properties in two cases: One is that N = 2 with subcritical exponential growth and critical exponential growth. The other is that N ≥ 3 under the indefinite case with two types of polynomial growth: super-quadratic and asymptotically quadratic. By using subtle estimates and establishing a new energy estimate inequality in complex field, we overcome the difficulty arising in strongly indefinite structure associated with the energy functional and the presence of magnetic field. Our results extend and complement the present ones in the literature. [ABSTRACT FROM AUTHOR]

We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equation with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ-translation invariance, and many effective methods for periodic problems cannot be applied to asymptotically periodic ones. These enables us to develop a direct approach to find ground state solutions with asymptotically periodic potential. Two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity which are weaker that some well-known ones. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear boundary value problem D D ψ - a (x) ψ = f (x , ψ) + ϵ h (x , ψ) on M B CHI ψ = 0 on ∂ M where M is a compact Riemannian spin manifold of dimension m ≥ 2 and the boundary ∂ M has non-negative mean curvature, and D is the Dirac-Atiyah-Singer operator. Let S(M) denote the spinor bundle on M and ψ : M → S (M) be a section. a(x) is a scalar field on M, ϵ ∈ R . f (x , ψ) , h (x , ψ) : M × S m → S m are two nonlinear map, and B CHI is the chirality boundary operator. Under some mild assumptions on a, f, and h, we obtain the ground state solution of (D) with ϵ = 0 and infinitely many large norm solutions of (D) with ϵ ∈ R and infinitely many small energy solutions of (D) with ϵ > 0 by using variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of K-component coupled nonlinear Schrödinger equations: - Δ u i + V i (x) u i = f i (x , u) in R N , i = 1 , 2 , … K , with u = (u 1 , u 2 , … , u K) : R N → R K , f i (x , u) = ∂ u i F (x , u) , which originates from Bose–Einstein condensates phenomenon. We mainly study the case that V i (x) is asymptotically periodic or non-periodic, which is different from periodic case. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of strongly indefinite structure. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Furthermore, we fill the gaps about the existence of ground state solutions of K-component equations with spectrum point zero. Nevertheless, we need to overcome some difficulties: one is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The second is that the working space is only a Banach space, not a Hilbert space, due to 0 is a boundary point of the spectrum of operator. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity, periodic assumption on potential and in verifying the link geometry and showing the boundedness of Cerami sequences. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |u(t, x)|+|v(t, x)|\rightarrow 0, \ \ \text{as}\ \ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $where $ 0

In this paper, we study the discrete nonlinear Choquard equation - Δ u + V (x) - ρ (| x | 2 + 1) u = (R α ∗ | u | p ) | u | p - 2 u , u ∈ ℓ 2 (Z N) , where N ≥ 3 , α ∈ (0 , N) , p > N + α N and R α is the Green’s function of the discrete fractional Laplacian, which has no singularity but has same asymptotics as the Riesz potential. We assume that V is a bounded periodic potential and 0 lies in a spectral gap of the discrete Schrödinger operator - Δ + V , which makes the functional is strongly indefinite. We prove the existence and asymptotic behavior of ground state solutions by the generalized linking theorem for small ρ ≥ 0 . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to look for solutions of the following nonlinear Maxwell system: where Ω ⊂ R 3 is a bounded domain with exterior normal ν : ∂ Ω → R 3 , ∇ × denotes the curl operator in R 3 , V 1 , V 2 : Ω → R are two continuous functions, μ 1 , μ 2 and λ are constants, 2 < p < 6 , α , β > 1 and α + β = p . By using some variational approach, we will show that there exists a ground state solution for system (M). Moreover, if Ω is a cylindrical symmetric domain, then system (M) has infinite bound state solutions with cylindrical symmetry. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of fractional Schrödinger equation with Hardy potential (− Δ) s u + V (x) u − κ | x | 2 s u = f (x , u) , x ∈ R N , where s ∈ (0 , 1) and κ ⩾ 0 is a parameter. Under some suitable conditions on the potential V and the nonlinearity f, we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ. Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of ground state solution for the following reaction-diffusion equation ∂ t u - Δ x u + b (t , x) · ∇ x u + V (x) u + λ v = g (t , x , v) , - ∂ t v - Δ x v - b (t , x) · ∇ x v + V (x) v + λ u = f (t , x , u) , where (t , x) ∈ R × R N , z = (u , v) : R × R N → R × R , b = (b 1 , b 2 ,... , b N) ∈ C 1 (R × R N) , V ∈ C (R N , R) , f , g ∈ C (R × R N × R , R) , all four are depending periodically on t and x. The resulting problem engages four major difficulties: one is that the associated functional is strongly indefinite, the second is that the working space for our case is only a Banach space, not a Hilbert space, due to 0 is a boundary point of the spectrum of operator. The third difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences. The last difficulty is due to the lack of the strict monotonicity condition, seeking a ground state solution on the Nehari-Pankov manifold is difficult which needs some new techniques and methods. These enable us to develop a direct approach and new tricks to overcome the above difficulties. We obtain ground state solutions of Nehari-Pankov type under weaker conditions on the nonlinearity based on non-Nehari manifold method developed recently. To our best knowledge, our theorems appear to be the first such results about ground state solutions of diffusion equations with spectrum point zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions. [ABSTRACT FROM AUTHOR]

This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following Hamiltonian elliptic system

Abstract In this paper, we study the second-order Hamiltonian systems u ¨ − L ( t ) u + ∇ W ( t , u ) = 0 , $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$ where t ∈ R $t\in \mathbb{R}$ , u ∈ R N $u\in \mathbb{R}^{N}$ , L and W depend periodically on t, 0 lies in a spectral gap of the operator − d 2 / d t 2 + L ( t ) $-d^{2}/dt^{2}+L(t)$ and W ( t , x ) $W(t,x)$ is locally superquadratic. Replacing the common superquadratic condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ uniformly in t ∈ R $t\in \mathbb{R}$ by the local condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ a.e. t ∈ J $t\in J$ for some open interval J ⊂ R $J\subset \mathbb{R}$ , we prove the existence of one nontrivial homoclinic soluiton for the above problem.

Abstract Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, q i ¨ = Φ i − 1 ′ ( q i − 1 − q i ) − Φ i ′ ( q i − q i + 1 ) , i ∈ Z , $$\begin{aligned} \ddot{q_{i}}=\varPhi _{i-1}'(q_{i-1}-q_{i})- \varPhi _{i}'(q_{i}-q_{i+1}),\quad i\in \mathbb{Z}, \end{aligned}$$ where q i $q_{i}$ denotes the co-ordinate of the ith particle and Φ i $\varPhi _{i}$ denotes the potential of the interaction between the ith and the ( i + 1 ) $(i+1)$ th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.

Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, q i ¨ = Φ i − 1 ′ (q i − 1 − q i) − Φ i ′ (q i − q i + 1) , i ∈ Z , where q i denotes the co-ordinate of the ith particle and Φ i denotes the potential of the interaction between the ith and the (i + 1) th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and concentration of ground state solution for the Choquard equation - Δ u + V (x) u = ∫ R N A (ϵ y) | u (y) | p | x - y | μ d y A (ϵ x) | u | p - 2 u in R N , u ∈ H 1 (R N) , where N ≥ 2 , 0 < μ < 2 , ϵ is a positive parameter. V is a Z N -periodic function, and 0 lies in a gap of the spectrum of - Δ + V . A ∈ C (R N) satisfies 0 < inf x ∈ R N A (x) ≤ lim | x | → + ∞ A (x) < sup x ∈ R N A (x). [ABSTRACT FROM AUTHOR]

In this paper, we consider the following semilinear elliptic systems: − Δ u + V (x) − μ | x 1 − x 2 | 2 | x − x 1 | 2 | x − x 2 | 2 u = F u (x , u , v) , in R N ∖ { x 1 , x 2 } , − Δ v + V (x) − μ | x 1 − x 2 | 2 | x − x 1 | 2 | x − x 2 | 2 v = F v (x , u , v) , in R N ∖ { x 1 , x 2 } , where x 1 ≠ x 2 , 0 ∉ σ (− Δ + V) , μ < ((N − 2) 2 / 4) , N ≥ 3 , V (x) , F u (x , u , v) and F v (x , u , v) are periodic in x. We impose general assumptions on the nonlinear term F with subcritical growth and superlinear condition. For sufficiently small μ ≥ 0 , we find a ground state solution being a minimizer of the energy functional associated with the above system on a Nehari-Pankov manifold. If μ < 0 and 0 < inf σ (− Δ + V) , then the ground state solutions do not exist. Recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to prove a fountain theorem without assuming the τ-upper semi-continuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

In this paper, we consider the following semilinear elliptic systems: - Δ u + V (x) u = F u (x , u , v) , in R N , - Δ v + V (x) v = F v (x , u , v) , in R N , where V : R N → R , F u (x , u , v) and F v (x , u , v) are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of - ▵ + V . Under appropriate assumptions on F u (x , u , v) and F v (x , u , v) , we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

Consider the second order self-adjoint discrete Hamiltonian system 6[p(n)6u(n - 1)] - L(n)u(n) + ∇W (n, u(n)) = 0, where p(n), L(n) and W (n, x) are N -periodic on n, and 0 lies in a gap of the spectrum σ(A) of the operator A, which is bounded self-adjoint in l2 (Z, RN ) defined by (Au)(n) = 6[p(n)6u(n - 1)] - L(n)u(n). We obtain a sufficient condition on the existence of nontrivial homoclinic orbits for the above system W (n,x) under a much weaker condition than lim…uniformly in n ∈ Z, which has been a common condition used in the existing literature. We also give three examples to illustrate our result. [ABSTRACT FROM AUTHOR]

This article concerns the Schr\"odinger equation $$\displaylines{ -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\cr u(x)\to 0, \quad \text{as } |x| \to \infty, }$$ where V and f are periodic in x, and 0 is a boundary point of the spectrum $\sigma(-\Delta+V)$. Assuming that f(x,u) is asymptotically linear as $|u|\to\infty$, existence of a ground state solution is established using some new techniques.

We survey recent results on ground and bound state solutions E:Ω→R3 of the problem {∇×(∇×E)+λE=|E|p−2EinΩ,v×E=0on∂Ω on a bounded Lipschitz domain Ω ⊂ ℝ3, where ∇× denotes the curl operator in ℝ3. The equation describes the propagation of the time-harmonic electric field R{E(x)eiωt} in a nonlinear isotropic material Ω with λ=−μεω2⩽0, where μ and ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term |E|p−2E with 2

We look for solutions E : Ω → R 3 of the problem { ∇ × ( ∇ × E ) + λ E = | E | p − 2 E in Ω ν × E = 0 on ∂ Ω on a bounded Lipschitz domain Ω ⊂ R 3 , where ∇× denotes the curl operator in R 3 . The equation describes the propagation of the time-harmonic electric field ℜ { E ( x ) e i ω t } in a nonlinear isotropic material Ω with λ = − μ ε ω 2 ≤ 0 , where μ and ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term | E | p − 2 E with p > 2 is responsible for the nonlinear polarisation of Ω and the boundary conditions are those for Ω surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical value p , for instance, in convex domains Ω or in domains with C 1 , 1 boundary, p = 6 = 2 ⁎ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on λ ≤ 0 . We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems. [ABSTRACT FROM AUTHOR]

This paper is focused on following time-harmonic Maxwell equation: [ABSTRACT FROM AUTHOR]

We study the Schrödinger equations –Δu + V (x)u = f(x, u) in RN and –Δu–λu = f(x, u) in a bounded domain Ω ⊂ RN. We assume that f is superlinear but of subcritical growth and u ⟼ f(x, u)/|u| is nondecreasing. In RN we also assume that V and f are periodic in x1, . . ., xN. We show that these equations have a ground state and that there exist infinitely many solutions if f is odd in u. Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802-3822], where u ⟼ f(x, u)/|u| was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis. [ABSTRACT FROM AUTHOR]

We consider a time-independent fractional Schrödinger equation ( − △ ) α u + V ( x ) u = f ( x , u ) in R N , u ∈ H α ( R N ) , where α ∈ ( 0 , 1 ) , N > 2 α , V ( x ) is a periodic potential, f is superlinear and has a general subcritical growth. Based on a generalized linking theorem and a variant fountain theorem for strongly indefinite functional, we obtain a ground state solution and infinitely many solutions. [ABSTRACT FROM AUTHOR]

We are concerned with a system of coupled Schrödinger equationswhereFandViare periodic inxand 0∉σ(−Δ+Vi) fori = 1,2,…,K, whereσ(−Δ+Vi) stands for the spectrum of the Schrödinger operator −Δ+Vi. We impose general assumptions on the nonlinearityFwith the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari–Pankov manifold. Our approach is based on a new linking-type result involving the Nehari–Pankov manifold. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following time-harmonic semilinear Maxwell equation: where $${\Omega\subset \mathbb{R}^{3}}$$ is a bounded, convex domain and $${\nu : \partial \Omega\to \mathbb{R}^{3}}$$ is the exterior normal. Motivated by recent work of Bartsch and Mederski and based on some observations and new techniques, we study above equation by developing the generalized Nehari manifold method. Particularly, existence of ground-state solutions of Nehari-Pankov type for the equation is established with asymptotically linear nonlinearity. [ABSTRACT FROM AUTHOR]

We study the existence of solutions of the following nonlinear Schrödinger equation − Δ u + ( V ( x ) − μ | x | 2 ) u = f ( x , u ) for x ∈ R N ∖ { 0 } , where V : R N → R and f : R N × R → R are periodic in x ∈ R N . We assume that 0 does not lie in the spectrum of − Δ + V and μ < ( N − 2 ) 2 4 , N ≥ 3 . The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small μ ≥ 0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If μ < 0 and 0 lies below the spectrum of − Δ + V , then ground state solutions do not exist. [ABSTRACT FROM AUTHOR]

In this paper, we construct a Rabinowitz–Floer type homology for a class of non-linear problems having a starshaped potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations. [ABSTRACT FROM AUTHOR]

We study the existence of homoclinic solutions for the following second-order self-adjoint discrete Hamiltonian system: $\triangle[p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0$, where $p(n)$, $L(n)$, and $W(n, x)$ are N-periodic in n, and $\nabla W(n, x)$ is asymptotically linear in x as $|x|\to\infty$. [ABSTRACT FROM AUTHOR]

A Based on a generalized linking theorem for the strongly indefinite functionals, we study the existence of homoclinic orbits of the second order self-adjoint discrete Hamiltonian system Δ[p(n)Δ(n - 1)] - L(n)u(n) + ∇W(n, u(n)) = 0, where p(n), L(n) and W(n, x) are N-periodic on n, and 0 lies in a gap of the spectrum σ(A) of the operator A, which is bounded self-adjoint in l² (ℤ, ℝN) defined by (Au)(n) = Δ[p(n)Δu(n-1)] - L(n)u(n). Under weak superquadratic conditions, we establish the existence of homoclinic orbits. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Choquard system in R N − Δ u + V (x) u = 2 q p + q ( I α ∗ | u | p ) | v | q − 2 v , − Δ v + V (x) v = 2 p p + q ( I α ∗ | v | q ) | u | p − 2 u , u (x) → 0 a n d v (x) → 0 a s | x | → ∞ , where α ∈ (0 , N) and N + α N < p , q < 2 ∗ α , in which 2 ∗ α denotes N + α N − 2 if N ≥ 3 and 2 ∗ α ≔ ∞ if N = 1 , 2. I α is a Riesz potential. V is continuous, 1-periodic in x and 0 belongs to a spectral gap of − Δ + V. This system is subcritical in the sense of the Hardy–Littlewood–Sobolev inequality. Based on the generalized Nehari manifold, we obtain the existence of ground state solutions. Our results can be looked on as a generalization to results by Szulkin and Weth (2009). [ABSTRACT FROM AUTHOR]

We consider a class of elliptic systems leading to strongly indefinite functionals, with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. Our approach relies on new critical point theorems which guarantee the existence of infinitely many critical values of a wide class of strongly indefinite even functionals. Our abstract critical points theorems generalize the fountain theorems of T. Bartsch and M. Willem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the existence of ground state solutions for Schrödinger equation where , V is periodic in x and 0 lies in a spectral gap of , K and f are asymptotically periodic in x. [Copyright &y& Elsevier]

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions. [ABSTRACT FROM AUTHOR]

Abstract: We study some basic analytical problems for nonlinear Dirac equations involving critical Sobolev exponents on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals defined on -spinors with critical growth. We prove the existence of a non-trivial solution for the Brezis–Nirenberg type problem when the dimension m of the manifold is larger than 3. We also prove a global compactness result for the associated Palais–Smale sequences and the regularity of -weak solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the following Hamiltonian elliptic system for . Existence and multiplicity of solutions are obtained for the systems with periodic or non-periodic potentials V via variational methods. [ABSTRACT FROM AUTHOR]

Abstract: We consider the nonlinear stationary Schrödinger equation in . Here f is a superlinear, subcritical nonlinearity, and we mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of . Inspired by previous work of Li et al. (2006) and Pankov (2005) , we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed. For odd f, we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the bifurcation problems for strongly indefinite functional via Morse theory. The generalized topological degree for a class of vector fields is defined. As applications, we study the bifurcation problems for Hamiltonian system and noncooperative elliptic system. [Copyright &y& Elsevier]

Abstract: We correct the statement and the proof of Proposition 9 in [D. Bonheure, M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, http://dx.doi.org/10.1016/j.anihpc.2008.06.002]. [Copyright &y& Elsevier]

Abstract: We prove that the elliptic system where Ω is a regular bounded domain of , and , admits an unbounded sequence of solutions , provided and . We also prove a generic multiplicity result for exponents in the open region bounded by the lines , and the critical hyperbola. [Copyright &y& Elsevier]