Infinitely Many Subharmonic Solutions For One-dimensional P-sublinear Laplacian Equation

p-Laplacian equation is an important model of differential equation from non-Newtonianfluid theory and nonlinear elasticity. In this article, We consider the existence of infinitely many subharmonic solutions with large amplitude or with small amplitude for one-dimensinal p-Laplacian equation: (|x^'|^p－2x^')^'＋g(t,x)=0 . whereg∈C(R×R, R) is 2π-periodic in time t , and is p-sublinear at the infinity (In Chapter2) in the sense (?) and is p-sublinear at the origin (In Chapter 3) inthe sense (?).The Pioncar(?)-Birkhoff theorem is the main tool of study for the existence of infinitelymany subharmonic solutions for one-dimensinal p-Laplacian equation. Thus the key step in the research is to construct a twist annulus in the phase-plane.For one-dimensinal p-Laplacian equation p-sublinear at the infinity, the main difficult is to find the inner boundary of the twist annulus. When the solution moves around the origin, it will arrives the origin. Then the angle of the solution moves will be not determined. Then, we change the old equation, in the neighborhood of (0,0), into a new plane Hamiltian system which (0,0) is the only solution for the Hamiltian system. By p-sublinear and sign condition ,we can find a increasly function which can control the inner boundary. Later the outer boundary can be found by p-sublinear. By using Pioncar(?)-Birkhoff theorem in the twist annulus bounded by above inner and outer boundary, we can obtian the existence of the fixed points for the Pioncare map which corresponding to the subharmonic solutions for the new Hamiltian system. The rotation estimations for the subharmonic solutions imply that these solutions just the solutions of the old equation. Another difficult in the study of p-Laplacian equation is how to prove the uniquence of the solution for the initial value problem. We use an approximation approach to overcome this difficult.For one-dimensinal p-Laplacian equation p-sublinear at the origin, if we want to prove the uniqueness of the solution for the initial value problem, we need to construct a prior annulus. Then choose a smooth function sequence g_ε(t,x) which approach g(t,x), and prove the twist property for every new system at the outer boundary and inner boundary of the prior annulus. Then obtain the existence of infinitely many subharmonic solutions with sufficiently small amplitude by Pioncar(?)-Birkhoff theorem. Finally we can obtain the subharmonic solutions of the original system by Arzela-Ascoli theorem.