【作者】 赵娜；

【导师】 钱定边；

【作者基本信息】 苏州大学 ， 基础数学， 2011， 硕士

【摘要】 本文讨论二阶非线性奇异φ-Laplace算子方程(φ(x’))’+f(t,x)=0的无穷多个周期解(次调和解)的存在性问题,其中φ:(-α,α)→R(0<α<+∞)是单调递增同胚,满足φ(0)=0.f(t,x)是关于时间t为2π-周期的连续函数,满足一定的符号条件.对于特殊的其典型的模型就是相对场中带电粒子的运动的描述.本文通过改造与所考虑方程等价的Hamilton系统的方法,在所给条件下构造环域,使Hamilton系统的Poincare映射的若干次迭代在环域上满足扭转条件.然后应用Poincare-Birkhoff扭转定理得到新系统的次调和解的存在性.并且我们可以构造不同的环域并不断提高迭代次数得到无穷多个次调和解的存在性.本文的结果是奇异φ-Laplace算子方程无穷多个周期解(次调和解)的存在性的第一个结果.更多还原

【Abstract】 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)→R (0<a<+∞) is an increasing homeomorphism, satisfyingφ(0)= 0; And f(t, x) is a continuous function,2π-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.更多还原