Infinitely Many Subharmonic Solutions For Nonlinear Second Order Equations With Singular Ñ„-Laplace

In this paper we deal with the existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian (φ(x'))'+f(t, x)=0, whereφ:(-α,α)â†'R (0<a<+∞) is an increasing homeomorphism, satisfyingφ(0)= 0; And f(t, x) is a continuous function,2Ï€-period with respect to time t. Also, we assume some sign condition on f.For special which is a particular model for mechanical dynamics of particle moving under relativistic effect.In this paper we modify the considered equation into a new plane Hamiltonian system and we can construct an annulus such that some iteration of the Poincare map for Hamiltonian system is twist on the annulus. By using Poincare-Birkhoff twist theorem, we can obtain the existence of the fixed points for the iteration of the Poincare map which corresponding to the subharmonic solutions for Hamiltonian system. The existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian then is proved by considering more and more iterations on different annuli.As we known, this is the first result on the existence of infinitely many subhar-monic solutions for nonlinear second order equation involving singularφ-Laplacian.