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

Qiong Meng; X. H. Tang

doi:10.3934/cpaa.2012.11.945

Article Contents

2012, Volume 11, Issue 3: 945-958. Doi: 10.3934/cpaa.2012.11.945

This issue Previous Article Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities Next Article Large time behavior for the full compressible magnetohydrodynamic flows

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

Qiong Meng^1, and
X. H. Tang^2,

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

Received: August 31, 2010

Revised: August 31, 2011

Published: April 2012

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Abstract

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.

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Mathematics Subject Classification: 34C25, 37B30, 37J45.

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