Multiple Solutions For Higher Order Elliptic System

In this paper, we discuss the existence of multiple solutions for a class of higher order elliptic system, which is on the basis of a mathematical model of the suspension bridge presented by A.C.Lazer and P.J.McKenna.By the variational method together sub-super solution, we extend the famous Ambrosetti-Prodi(shortly named A-P)type results for elliptic equation to the second and fourth order ordinary differential system, and verify that there exists a continuous curve splitting the plane into two unbounded components, and the multiplicity and nonexistence of solutions when parameter t = (t 1 , t2) belong to the two components, respectively.Moreover, we generalize the second and fourth order ordinary differential system to higher dimension and obtain a class of second and fourth order elliptic system with variational structure. Applying the abstract linking theorem on product space, and by Nehari manifold, we study the existence of three nontrivial solutions of the system. Moreover, by the similar method and maximum principle, we discuss the existence of three nonnegative solutions under other conditions for the second and fourth order elliptic system.