Times The Second Critical Point Theory-based Fourth-order Linear Ordinary Differential Equations Periodic Solution Existence

In recent years there has been a considerable interest in fourth-order model equations such as the extended Fisher-Kolmogorov equation proposed in1988by Dee and Van Saarloos as a model for bi-stable systems, and the Swift-Hohenberg equation by Swift and Hohenberg in studies of hydrodynamic instability. After an appropriate transformation, standing wave solutions of these equations lead to the equation in which p>0corresponds to EFK equation and p<0to the SH equation. Many profound results about many other types of fourth-order nonlinear differential equations have been obtained in the use of the methods of the critical point theory developed in recent decades.To be more specific, we shall mainly discuss the existence of2T-periodic solutions for more general forth-order semi-linear differential equations Where A,B are constants, V(t,u)∈C1([O,T]×R,R) satisfies2V(t,u)-uVu(t,u)â†'∞,|u|â†'∞,t∈[O,T],or2V(t,u)â†'-∞,|u|â†'∞,t∈[O,T].Firstly, we will consider the following boundary problem If u is a solution of the above problem, since is f an even2T-periodic function with respect to x and odd with respect to u, then the IT-periodic extensionu=u(t) of the odd extension of the solution u to the interval [-T, T] is a2T-periodic solution of (1) on R.Problem (P) has a variational structure and its solutions can be found as critical points of the functional In the Sobolev space It is easy to prove that the critical point of I(u;T) is the classical solution of the boundary value problem (P). So, in this paper we mainly obtain nontrivial critical points of the functional/using different variational means.The paper contains three chapters. Chapter I is an introduction of research background and cents; Chapter II is preliminary knowledge which introduces the basic theory of critical points; In Chapter III, we mainly obtain nontrivial critical points of the functional/using different variational means such as the Mountain-pass theorem, the minimizing methods, the local linking theorem due to Brezis and Nirenberg, the Silva-linking theorem as A,B respectively satisfies corresponding conditions.