Resonant Phenomena Of 1-dimensional P-Laplacian Equation

p-Laplacian equation is an important model of differential equation from non- Newtonian fluid theory and nonlinear elasticity.In this paper,we will investigate the resonant phenomena of 1-dimensional p-Laplacian equation.Based on a generalized version of the Poincar(?)-Birkhoff twist theorem by J.Franks and W.Ding,we prove the existence of infinite subharmonic solutions for the equation.Moreover,we establish the coexistence of periodic solution and unbounded solution for the equation.From the view-point of a perturbation of the isochronous Hamiltonian system in the plane,the isochronous part for p-Laplacian equation is non-homogeneous in the variables. So we introduce some new action-angle variables to overcome this difficult.In the case of the perturbed term being bounded,we prove that Poincar(?) map of the equation has a twist property on some annulus.In the case of the perturbed term being unbounded,we don't know whether there is a twist property of Poincar(?) map or not.So we consider the successor map in the new action-angle coordinates and prove the twist property of the successor map. Then we obtain the existence of infinitely many periodic solutions for p-Laplacian equation by using Poincar(?)-Birkhoff twist theorem in both cases.When the perturbed term is small enough,we prove that all of solutions with large amplitude are unbounded under some conditions.Moreover,we also prove the existence of 2Ï€periodic solution for p-Laplacian equation by using Topological degree.At last,we give the condition for the coexistence of periodic solution and unbounded solution for the equation.