The Applications Of An Infinite Dimensional Morse Theory To A Class Of Asymptotically Linear Ordinary Differential Equation

The variational methods derived from the isopoerimetric problem in geometry and the problem of brachistochrone proposed by J Bernoulli in 17 century,these problem can be changed into a problem of seeking the critical point of a functional. In 18 century Euler and Lagrange gave a general method to solve them,and established Calculus of Variations which is an important branch of math.The basic idea of variational methods is changing the problem of seeking the critical point of a functional to be the problem of seeking the solution of an equation,and this equation is called the Euler-Lagrange equation of the functional.The solution of equation is the critical point of the functional,under some further condition it is the extremal function that we wanted.The isopoerimetric problem and the problem of brachistochrone can be solved by this path.The modern variational methods is changing the study of the solution of an equation to be the study of the critical point of a functional,such as the problem of periodic solution of Hamiltonian equation, the boundary value problem of semilinear elliptic equation and the periodic solution problem of semilinear wave equation which are well known.When the func-tional is bounded,we can use the method of minimizing sequences.But when the functional is not bounded,this method is not valid,and we need a new method.Now what we mainly relied is the tool of topology,so this part critical point theory is called Calculus of Variations in Large,in which Morse theory is representative.It was estab-lished by American mathematician Morse when he studied the geodesic problem of compact manifold. Morse showed the relation between the behaviour of the critical points of a Morse function which only have nondegenerate critical points and the topological structure of the manifold in a profound inequality, and so began the study of global analysis.This make us to use the topology method in the problem of analysis, but there is a difficulty when we want to apply the Morse theory to the differential equation.The old Morse theory always deal with the space of finite dimension,but the functional of the differential equation is in a function space of infinite dimension. So we need extend the Morse theory to the space of infinite dimension. In 1963,Palais and Smale extended the Morse theory to the Hilbert manifold of infinite dimension,and used it to study the nonlinear Dirichlet problem.In 1978,Conley also extended the Morse theory, he definite the Conley index for a class of isolated invariant set and established the Conley index theory for a gradient-like flow.In 1980, Amann and Zehnder use the Conley theory to the study of nontrivial solution of asymptotically linear operator equation, and obtained some important results in the periodic problem of Hamiltonian equation,the boundary value problem of elliptic equation and the periodic problem of wave equation.In 1981,Chang directly used the Morse theory to the problem of nontrivial solution of asymptotically linear operator equation,and obtained some further results. A lot of work began after Chang's work.The Morse index of the critical points of functional relate to the Hamiltonian equation and wave equation are all infinite, so we cannot directly use the Morse theory but use a saddle point reduce method which like the Lyapunov-Schmit reduce method to reduce the problem to a space of finite dimension. And this need a stronger restriction.In 1997, Kryszewski and Szulkin found a new cohomology in the space of infinite di-mension and based on it they established an infinite dimension Morse theory. This theory have been used in a lot of indefinite variational problem such as asymptotically linear Hamil-tonian equation, wave equation and beam equation.In the second chapter of this paper we introduce the theory established by Kryszewski and Szulkin,the detail may be found in their paper. First we definite an additional structure in the Hilbert space E.Suppose En is an filtration of E, and let every En to a nonnegative number dn, letε:={En,dn}n=1∞. Suppose (X,A) is a closed set pair of E, for every q, we definite the qthεcohomology group with coefficients in F:Then we definite the most important definition critical group at a isolated critical point. It is via a so called admissible pair which is like Gromoll Meyer pair, we always denote it (W, W-).Definition 1 Supposeε:={En, dn}n=1∞has been given,Φ∈C1 (E, R) satisfies (PS)* condition, p is an isolated critical point ofΦ,(W, W) is an admissible pair ofΦand p, We definite the qth critical group related toεofΦas:This new critical group do not depend on the choice of admissible pair,and it has the continuation property.Then we stated the Morse inequality which relate the local and global structure.Then we study a class of special operatorL=L-B:E→E,L:E→E is a linear adjoint Fredholm operator with zero index, and satisfy:L(En)(?)En, and M-(L|En=dn+k expect a finite n where M-is Morse index in the classical Morse theory, k is a constant chosen according to the problem. B:E→E is a linear compact adjoint operator. We definite itsεMorse index.Definition 2 Suppose Qn:R(L)→R(L)∩En is an projective operator from R(L) to R(L)∩En,εMorse index of the operator L is:For a class of functionalΦ(χ)=1/2-ψ(χ), we make use ofε-Morse index of L to compute or estimate its critical group at isolated critical point zero Cε*(Φ,0).At the first section of the third chapter we apply this theory to a class of asymptotically linear ordinary differential equation: where F∈C1(R x RN×RN, R) is a 2πperiodic function with respect to t. Let then we can rewrite the equation into a simper formulationFirstly, we give some assumption to this equation:(H1):Fz(t,0)=0,(?)t∈R(H2):Fz(t,z)=A0(t)z+(H0)z(t,z)(H0)z(t,z)=o(|z|) uniformly in t as|z|→0. Fz(t,z)=A(t)z+Hz(t,z)Hz(t,z)=o(|z|) uniformly in t as|z|→∞.(H3):F∈C2(R×R2N,R)and for someC＞0,s∈(0,∞)and all(t,z),|Fzz(t,z)|≤C(1＋|z|s)According to (H1) we see z=0 is a periodic solution of the equation, we call it trivial periodic solution,condition (H2) is called asymptotically linear condition. The equation is asymptotically linear near zero and infinity. In this chapter we shall consider the existence and multiplicity problem of this equation.Here we consider the Sobolev space EThe element of this space is which satisfy The inner product in the space isThe functional relate to this equation is Here L0 and L are the asymptotically operator at zero and infinity. LetHere is the image of the operator L,N(L) is the kernel of the operator L.For this functional we compute itsε-critical group at zero and infinity, and then obtain the existence and multiplicity result via compare them. The main results of this paper are:Theorem 1 SupposeΦsatisfies (H1),(H2),(H3).0 is a isolated critical point ofΦ, when one of the following condition(A1):M0(L0)=M0(L)=0,Mε-(L0)≠Mε-(L).(A2):Hz is bounded, H(t,z)→∞uniformly in t, as|Pz|→∞, and(A3):Hz is bounded, H(t,z)→∞, uniformly in t, as|z|→∞, and is satisfied,the functionΦhave at least one nontrivial critical point and then the equation have at least one nontrivial periodic solution.Theorem 2 SupposeΦsatisfies (H1),(H2),(H3).0 is a isolated critical point ofΦ, M0(L0)=0, when one of the following condition(A4):M0(L)=0,|Mε-(L0)-Mε--(L)|≥4N.(A5):Hz is bounded H(t, z)→∞uniformly in t as|z|→∞, and(A6):Hz is bounded, H(t,z)→-∞uniformly in t as|z|→∞, and is satisfied, the functionΦhave at least two nontrivial critical points and then the equation have at least two nontrivial periodic solutions.Finally, we consider an example: For this equation we can compute the e-Morse index of its functional at zero and infinity ,and hence give theε-critical group of the functional at zero and infinity. After compare them we can obtain the existence and multiplicity of the nontrivial periodic solution.Whenλ＞1/2,no matter zero is nondegenerate or not, we shall obtain the existence of nontrivial critical point according to theorem 1,so the equation have at least one nontrivial periodic solution.whenλ＞2, and for every integer m>1,λ≠2/m2,zero is a nondegenerate critical point, we can obtain another nontrivial critical point according to theorem 2, so the equation have at least two nontrivial periodic solutions.