Periodic Solutions Of Completely Resonant Nonlinear Beam Equation With General Nonlinearities

Dynamical system is an important component of the nonlinear science. It is a subject which takes research on practical problems of dynamics laws about the state changing with the time. At present, dynamical system has been taken widely applications on many natural science fields like classical mechanics, engineering design, signal transmission, astronomy, geography, meteorology, biology, information and computing sciences and social science fields and so on.Beam vibration phenomena is governed by a partial differential equation known as the beam equation. Such equation involves functions depending on one time variable and a given number of spatial variables. This partial differential equations (PDEs) arising in physics can be seen as Infinite Dimensional Hamiltonian Systemswhere Hamiltonian functionsis defined on an infinite dimensional Hilbert(or Banach) space H, J is a non-degenerate antisymmetric operator.For a long time, the dynamics of the solutions of the infinite dimensional dynamical systems has drawn the attention of many researchers, specifically the resonant situations or partially resonant situations are the hot issues to considered. Recent years, many researchersstudied Infinite Hamiltionian Dimensional Partial Differential Equations, such as Kdv equations [35], Boussinesq equations[9], nonlinear wave equations [1, 3, 4, 6, 8], and these equations have a strong physical background, the study of them will have a practical significance.For finite dimensional hamiltonian system, the importance of periodic solutions in order to understand the dynamics, was first highlighted by Poincar(?). He wrote (talking about the three body problem): "D'alleurs, ce qui nous rend ces solutions periodiques si pr(?) cieuses, c'est qu'elles sont, pour ainsi dire, la seule breche par o(?) nous puissons essayer de p(?)n(?)trer dans une place jusqui'ci reputee inabordable" . From now on many researchers start to research the periodic obit of the Hamiltonian systems, such as Weinstein[12], Bikhoff[10], Rabinowitz[11].In the early 90s, the local bifurcation theory of periodic (and quasi-periodic) solutions started to be extended to Hamiltonian PDEs by Kuksin, Wayne, Craig, Bourgain, Poeschel and followed by many others, they further develop the infinite dimensional dynamical systems.Take the ways Kuksin [39] and Wayne[22] they deal with, first: it allows one to construct solutions whose periods are irrational multiples ofÏ€and second it easily extends to give quasi-periodic as well as periodic solutions. A disadvantage is that since the KAM theory has an essentially perturbative character, it is restricted to equations with weak nonlinearityor equivalently to solutions of small norm. In 93, Craig[8] First introduce the Lyapunov-Schmidt reduction methods and Nash-Moser iteration for Hamiltonian PDEs to handled some resonant cases, and effective to overcome the small divisor problem. So we first explain the divisor problem, take the nonlinear beam equations in this paper for example, consider the following beam equationsNote that eigenvalues of the operator (?) in the spaces of functions u(t, x), (2Ï€/ω)-periodicin time and such that, say, u(t,·)∈H₀²(Ω) for all t , are -ω²l² + j⁴ (?)l∈Z,j≥1 Therefore, for almost everyω∈R, the eigenvalues accumulate to zero. As a consequence, for mostω, the inverse operator of is (?) unbounded, and the standard implicit function theorem is not applicable.In 2001, Bambusi-Paleari [14] consider the nonlinearity f(x, u) = u³, and get a family of periodic solutions of one class of resonant PDEs. To overcome the divisor problem, they imposed on the frequencyωthe strongly nonresonance condition Forγ< 1/3, the set W_γis uncountable, has zero measure and accumulates toω= 1 both from the left and from the right.For infinite dimensional Hamiltion PDEs, except the small divisor problem, solving the infinite dimensional bifurcation equation is also a difficulty.In 2003, M. Berti[3] was to solve the bifurcation equation via a variational principle at fixed frequency which, jointly with min-max arguments, to find solutions of nonlinear wave equation as critical points of the Lagrangian action functional. More precisely, the bifurcation equation is, for any fixedω∈R , the Euler-Lagrange equation of a reduced Lagrangian action functional which possesses nontrivial critical points of mountain pass type.On the other infinite dimensional Hamiltonian PDEs-beam vibration model of the discussion,in recent years, a lot of mathematicians in-depth study and carried out by a series of interesting results, such as :In 2000 and 2001, Bamusi[14], Bambusi and Paleari[23], They found that an approach based on Lyapunov-Schmidt reduction to construct One-dimensional wave equation and the periodic solutions for beam equations.In 2006, Zhenguo Liang and Jiansheng Geng[24], they used the infinite dimensional KAM theorem, partial normal form and scaling skills and get the quasi-periodic solutions of the following beam equationsIn 2007, Huawei Niu and Jiansheng Geng [21] improved the infinite dimensional KAM Theorem, get almost periodic solutions for a family of high-dimensional beam equation.We mainly use the methods in literature [3] given the variational principle and the combinationof Lyapunov-Schmidt reduction to get the periodic solutions of the the beam equationwith forced vibration, and the nonlinear terms start from the second-order. Cleverly in order to avoid the small divisor problem, similar to the literature [3] and [14] add the non-resonanceconditions, In line with the beams together with the corresponding vibration of the non-resonance. In the first section, we will introduce the the filed of background and some relevant results about the Infinite Dimensional Hamiltonian Systems. In the second section, we will introduce the mountain pass lemma and variational Lyapunov-Schmidt reduction. In the third section, by using the technique and method in literature [3], we will give the periodic solutions of the completely resonant beam equations.We consider the following one-dimensional nonautonomic completely resonant beam equation:on the finite x-interval [0,Ï€] with Dirichlet boundary conditionsForcing term f(t,x, u) is assumed as followswe use a variational principle given by Berti and Bolle [3] for getting solutions with fixed frequency: we impose the frequency to satisfy a corresponding strong irrationality property for overcoming the small divisor problem and we solve problem infinite dimensionalbifurcation equations exploiting min-max variational arguments. More precisely, by the variational Lyapunov-Schmidt procedure one has to solve two equations: the (?) equationwhere the small denominators problem appears, and the (Q) equation which is the bifurcation equation on the infinite dimensional space of solutionsThe (P) equation is solved by an Implicit Function Theorem. In order to focus our attention on problem of (Q) equation, we simplify the small denominators problem of (Q) equation imposing on the frequencyωthe strong irrationality condition We prove this set satisfy the property of asymptotically full measure, see the following LemmaLemma 1 Let (ω,ε)∈(1 -ω₁, 1 +ω₁)×(0,η), and there existsγ> 0 such thatthe Lebesque measure of the set W_γis asymptotically full, i.e.meas(W_γ)≥η(2ω₁ - C_γ).Once the (P) equation is solved, it remains the infinite dimensional (Q) equation: we solve it via a variational principle, noting that it is the Euler-Lagrange equation of a reduced action functionalΦ_ωdefined in (2.1) like in [3], [4]. Non-trivial critical points ofΦ_ωcan be obtained by min-max variational arguments. So The main results of this paper as followsCaseâ… p is odd, p≥3.Theorem 1 Let f(t, x,u):= (?) a_p+k(t, x)u^p+k and p is odd integer, a_p(t, x) > 0, (?)(t, x)∈Ω. there exists positive number C := C(f), such that (?)ω∈W_γsatisfying |ω- 1|≤C, equation (1.4)-(1.5) possesses at least one non-trivial periodic solution with 2Ï€/ω, and the solution u_ωconverges to zero lim(ω→1) ||u_ω|| = 0.Caseâ…¡p is even, p≥2.Theorem 2 Let f(t, x,u):= (?) a_p+k(t, x)u^p+k, a_p(t, x) satisfying a_p(t+Ï€,Ï€-x) - a_p(t, x)and there exists v∈V such that (?) a_p(t, x)v^pL^-1a_p(t, x)v^p≠0. Then there exists positive number C := C(f), such that (?)ω∈W_γsatisfying |ω- 1|≤C, equation (1.4)-(1.5) has at least one non-trivial periodic solution with 2Ï€/ω, and the solution u_ωconverges to zero lim_{ω→1} ||u_ω|| = 0.