American Institute of Mathematical Sciences

May  2012, 11(3): 945-958. doi: 10.3934/cpaa.2012.11.945

Multiple solutions of second-order ordinary differential equation via Morse theory

 1 School of Mathematics Sciences, Shanxi University, Taiyuan, Shanxi 030006, China 2 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083

Received  September 2010 Revised  September 2011 Published  December 2011

In this paper, we consider the the second-order ordinary differential equation with periodic boundary problem $- \ddot{x}(t)=f(t,x(t))$, subject to $x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0$, where $f:C([0, 2\pi]\times R, R)$. The operator $K=(-\frac{d^2}{dt^2}+I)^{-1}$ plays an important role. By using Morse index, Leray-Schauder degree and Morse index theorem of the type Lazer-Solimini, we obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.
Citation: Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
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