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.

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.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

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 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).

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively.

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.

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.

The New Infinite Sequence Solutions of Multiple Sine-Gordon Equations

By the function transformation and the first integral of the ordinary differential equations, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is researched, and the new solutions are obtained. First, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is changed to the problem of solving the solutions of the nonlinear ordinary differential equation. Second, with the help of the B?cklund transformation and the nonlinear superposition formula of solutions of the first kind of elliptic equation and the Riccati equation, the new infinite sequence soliton-like solutions of two kinds of sine-Gordon equations are constructed.

THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

This paper considers the following quasilinear elliptic problem [GRAPHICS] where Omega is a bounded regular domain in R-N (N greater than or equal to 3), N > p > 1. When g(u) satisfies suitable conditions and g(u)u - beta integral (u)(0) g(s)ds is unbounded, a(x) is a Holder continuous function which changes sign on Omega and integral (Omega-) \a(x)\ dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N > p > 1. in particular, the existence of a positive solution to the problem for N > p greater than or equal to 2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2. They prove also that if g(u) = \u \ (q-2)u with p < q < p* and Omega (+) = {x is an element ofQ \a(x) > 0} is a nonempty open set, then the above problem possesses infinitely many solutions.

ANALYTICAL SOLUTION OF FLOW IN INFINITE STRATIFIED OIL RESERVOIR AND ITS APPLICATION

This paper presents an analytical solution for the production function and pressure distribution function of flow in infinite stratified oil reservoir with crosflow under the condition of constant wellbore pressure (CWP condition) by Weber's integral transformation. The calculation results are shown in the form of curves and these results can be used to analyse unsteady flow test of production with CWP condition.

THE SINGULAR SECOND ORDER NONLINEAR EIGENVALUE PROBLEM WITH INFINITELY MANY POSITIVE SOLUTIONS

In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.

FOUNTAIN THEOREM OVER CONES AND APPLICATIONS

In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem{-△Pu=λ｜u｜q-2u＋μ｜u｜ y-2u,x∈Ω,u=0,x∈δΩ to show that problem （1） possesses infinitely many solutions, where 1 〈 p 〈 N, 1 〈 q 〈 P 〈 γ, Ω∩→ R^N is a smooth bounded domain and λ, μ∈ R.

Infinitely Many Solutions for the Fractional Nonlinear Schrodinger Equations ofa New Type

This paper,we study the multiplicity of solutions for the fractional Schrodingerequation(-△)^(s)u+V(x)u=u^(p),u>0,x∈R^(N),u∈H^(s)(R^(N)),with s∈(0,1),N≥3,p∈(1,2N/N-2s-1)and lim_(|y|→+∞)V(y)>0.By assuming suitable decay property of the radial potential V(y)=V(|y|),we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides.Hence,besides the length of each polygonal,we must consider one more parameter,that is the height of the podetium,simultaneously.Another difficulty lies in the non-local property of the operator(-△)^(s) and the algebraic decay involving the approximation solutions make the estimates become more subtle.

Infinitely Many Solutions for an Elliptic Problem with Critical Exponent in Exterior Domain

We consider the following nonlinear problem {-△u=uN＋2/N-2,u〉0 in R^N/Ω,u（x）→0 as｜x｜→＋∞,δu/δn=0 on δΩ,where Ω belong to RN,N ≥ 4 is a smooth and bounded domain and n denotes inward normal vector of δΩ. We prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large when Ω is convex seen from inside （with some symmetries）.

ON PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,using Mawhin's continuation theorem in the theory of coincidence degree,we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay:dx(t)/dt=f(t,x_t),x(t)∈R^n,which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay.Second,as an application of it,we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations:dx(t)/dt=x(t)(a-kx(t)-by(t)),dy(t)/dt=-cy(t)+d integral from n=0 to +∞ x(t-s)y(t-s)dμ(s)+p(t).

Divergent Solutions to the L^2-Supercritical NLS Equations Qing GUO

We investigate the nonlinear Schrdinger equation iut＋△u＋｜u｜^p-1u = 0 with 1＋4/N 〈 p 〈 1＋4/（N-2）（when N = 1,2,1 ＋4/N 〈 p 〈 ∞） in energy space H^1 and study the divergent property of infinite-variance and nonradial solutions.If M（u）^（1-sc）/sc E（u） 〈 M（Q）^（1-sc）/scE（Q） and ｜｜u0｜｜0^（1-sc）/sc ｜｜▽u0｜｜2 〉 ｜｜Q｜｜^（1-sc）/sc ｜▽Q｜｜2,then either u（t） blows up in finite forward time or u（t） exists globally for positive time and there exists a time sequence tn→ ＋∞ such that ｜｜ ▽u（tn）｜｜2 →＋∞.Here Q is the ground state solution of —（1 — sc）Q ＋ △Q ＋ ｜Q｜p-1Q = 0.A similar result holds for negative time.This extend the result of the 3D cubic Schrodinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).

EXISTENCE AND MULTIPLICITY RESULTS FOR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV EXPONENT AND HARDY TERM

This paper concerns the existence and multiplicity of solutions for some semilinear elliptic equations with critical Sobolev exponent, Hardy term and the sublinear nonlinearity at origin. By using Ekeland,s variational principle, we conclude the existence of nontrivial solution for this problem, the Clark＇s critical point theorem is used to prove the existence of infinitely many solutions for this problem with odd nonlinearity.