Periodic Solutions And Ground States Of The Fourth-order Differential Equations

This paper is mainly composed of three chapters:Chapter 1 is introduction. Chapter 2 studies the existence and multiplicity of T-periodic solutions for a class of fourth-order Hamiltonian systems by the critical point theory. In Chapter 3, we study the existence of ground states for some fourth-order periodic boundary value problems with Ambrosetti and Rabinowiz growth condition, applying a minimax characterization of ground states.Consider the fourth-order Hamiltonian systems where A(t),B(t) are N x N symmetric matrixes, A(·) continuous, B(·) continuously differ-entiable, T-periodic, and is T-periodic in t, T> 0, F(t,0)= 0, t∈R, and satisfies the following assumption:(A) F(·,x) is measurable in t for every x∈RN,F(t,·) is continuously differentiable in x for a.e. t∈[0,T], and there exist a∈C(R+,R+),b∈L1([0,T],R+) such thatThe following theorems are the main results of this paper:Theorem2.1 Suppose that F satisfies (A) and the following conditions:(F3) There existsλ> 2, d1> 0 such that uniformly for a.e. t∈[0, T];(F4) There existsβ>λ-1, d2> 0 such that d2, uniformly for a.e. t∈[0,T];If 0 is an eigenvalue of with periodic boundary condition, then problem (2.1) has at least one nontrivial T-periodic solution.Theorem2.2 Suppose that F(t,·) is even in x for a.e. t∈[0,T], i.e. F(t,-x)= F(t,x),x∈RN,a.e. t∈[0,T], and satisfies the all conditions of the Theorem2.1, then problem (2.1) has infinitely many T-periodic solutions. Consider the fourth-order periodic boundary value problems: where f:[0, T]×R→R is continuous. Suppose that f satisfies the following conditions:(f1) There exists C0> 0 such that|f(t,x)|≤C0(|x|+|x|p-1),(t,x)∈[0,T]×R, where p>2;(f2) f(t,x)= o(x),x→0, uniformly for t∈[0,T];(f3) There existsα> 2 such that aF(t, x)≤xf(t, x), (t, x)∈[0, T]×R, where F(t,x)=∫0x f(t,y)dy;(f4) There exists R> 0 such that(f5) f(t,x)/|x| is strictly increasing in x for all t∈[0, T].There obtains the following theorem:Theorem3.1 If (f1)-(f5) hold, then problem (3.1) has a ground state in C4[0,T].