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Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

Authors:	Email author" target="_blank">Zi-qing?Xie Email author Chuan-miao?Chen

Institution:	(1) Institute of Computation, Hunan Normal University, Changsha, Hunan, China

Abstract:	Abstract The multiple solutions for one-dimensional cubic nonlinear problem u″ + u³ = 0, u(0) = u(π) = 0 are computed, on the basis of the eigenpairs of $- {\phi }\ifmmode{'}\else$'$\fi_{k} = \lambda _{k} \phi _{k}$ , k = 1, 2, 3 · · ·. There exist two nonzero solutions ±u _k corresponding to each k, and their Morse index MI(k) for 1 ≤ k ≤ 20 is to be exactly determined. It is shown by the numerical results that MI(k) ≥ k. Supported by The Special Funds of State Major Basic Research Projects (No.G1999032804), National Natural Science Foundation of China (No.19331021) and Mathematical Tianyuan Youth Foundation of National Natural Science Foundation of China (No.10226016).

Keywords:	Nonlinear problem multiple solutions series expression Morse index
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