Infinitely Many Solutions and a Ground-State Solution for Klein-Gordon Equation Coupled with Born-Infeld Theory

In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

INFINITELY MANY SOLUTIONS FOR AN ELLIPTIC PROBLEM INVOLVING CRITICAL NONLINEARITY

We study the following elliptic problem：{-div（a（x）Du）=Q（x）｜u｜2-2u＋λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.

POSITIVE SOLUTIONS AND INFINITELY MANY SOLUTIONS FOR A WEAKLY COUPLED SYSTEM

We study a Schrodinger system with the sum of linear and nonlinear couplings.Applying index theory,we obtain infinitely many solutions for the system with periodic potent ials.Moreover,by using the concentration compactness met hod,we prove the exis tence and nonexistence of ground state solutions for the system with close-to-periodic potentials.

Infinitely many periodic solutions for second-order Hamiltonian systems

The existence of high energy periodic solutions for the second-order Hamiltonian system -ü（t）＋A（t）u（t）=▽F（t,u（t）） with convex and concave nonlinearities is studied, where F（t, u） = F1（t,u）＋F2（t,u）. Under the condition that F is an even functional, infinitely many solutions for it are obtained by the variant fountain theorem. The result is a complement for some known ones in the critical point theory.

INFINITELY MANY SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).

INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

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 -△u-μ（u/｜x｜^2）=λu＋｜u｜^2＊-2u inΩ, u=0 on eΩ,where Ω is a smooth open bounded domain of R^N which contains the origin, 2＊=2N/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 Lj usternik-Schnirelman theory.

INFINITELY MANY SOLUTIONS FOR A SINGULAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS IN R^N

In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β.

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT

This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical exponent for the Sobolev imbedding, λ > 0 and f(x, u) satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p = 2 or f(x,u) = |u|q-2u, where 1 < q < p, are generalized.

INFINITELY MANY SOLUTIONS WITH PEAKS FOR A FRACTIONAL SYSTEM IN R^N

In this article,we consider the following coupled fractional nonlinear Schrödinger system in R^{(−Δ)su+P(x)u=μ1|u|^2p−2u+β|u|p|u|p−2u,x∈RN,(−Δ)sv+Q(x)v=μ2|v|^2p−2v+β|v|p|v|p−2v,x∈RN,u,v∈Hs(RN),where N≥2,0<s<1,1<p<NN−2s,μ1>0,μ2>0 andβ∈R is a coupling constant.We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x),Q(x),p andβ.More precisely,we will show that for the attractive case,it has infinitely many non-radial positive synchronized vector solutions,and for the repulsive case,infinitely many non-radial positive segregated vector solutions can be found,where we assume that P(x)and Q(x)satisfy some algebraic decay at infinity.

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R³

In this paper, we consider the following fourth-order equation of Kirchhoff type where a, b > 0 are constants, 3 < p < 5, V ∈ C (R3, R);Δ2: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on V (x). We make some assumptions on the potential V (x) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature.

Interaction between infinitely many dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal

By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions of stress fields of the interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are obtained. They indicate that the stress concentration occurs at the dislocation source and the tip of the crack, and the value of the stress increases with the number of the dislocations increasing. These results are the development of interaction among the finitely many defects of quasicrystals, which possesses an important reference value for studying the interaction problems of infinitely many defects in fracture mechanics of quasicrystal.

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.

INFINITELY MANY SOLITARY WAVES DUE TO THE SECOND-HARMONIC GENERATION IN QUADRATIC MEDIA

In this paper, we consider the following coupled Schrodinger system with χ^(2) nonlinearities {-Δu1+V1(x)u1=αu1u2,x∈R^N,-Δu2+V2(x)u2=α/2u^21+βu^22,x∈R^N.which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N < 6, α > 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.

THE EXISTENCE OF INFINITELY MANY SO UTIONS OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS

In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.

EXISTENCE,UNIQUENESS AND APPROXIMATION OF THE SOLUTION OF AN ODE WITH (INFINITELY MANY) STATE-DEPENDENT IMPULSES VIA FIXED MESH GALERKIN FORMULATION

A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.

Countably Many Positive Solutions for Nonlinear Singular <i>n</i>-Point Boundary Value Problems

In this paper, a fixed-point theorem has been used to investigate the existence of countable positive solutions of n-point boundary value problem. As an application, we also give an example to demonstrate our results.

Infinitely Many Symmetries of Konopelchenko-Dubrovsky Equation

A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Q to (F(x,u,Du))+integral from n=■Q to (G(x,u))

In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.