Abstract
本篇論文研究下列微分方程式的解的個數問題。-u''= .lambda.(.beta.-u)^p exp(-.gamma./(1+u)), 0<x<1, u'(0)=u(1)=0, 其中 .lambda.,.beta., .gamma.>0 且 p .gdsim. 0。此一微分方程式為催化理論中一個重要微分方程式的一維情形。f(u)=(.beta.-u)^p exp(-.gamma./(1+u)) 表示 p 級反應速率, u 表示溫度, .beta.表示熱效應,.gamma.表示催化能。本文中將導出在 p .gdsim. 0 時,對於所有的 .lambda.>0,此一微分方程式僅有一個解的明確條件。另外,本文中也將導出在 0.ldsim. p .ldsiam.1 時, 此一微分方程式的分歧曲線呈現 "S" 形的明確條件。We study the number of solutions to the equation -u" =.lambda.(.beta.-u)^p exp((-.gamma.)/(1+u)), 0 < x < 1, with the boundaryconditions u'(0) = u(1) = 0, where the parameters .lambda.,.beta., .gamma. > 0 and p >= 0. This is one-dimensional caseof an equation of some importance in catalysis theory. Wegive explicit criterions for a unique solution for all .lambda.> 0 when p >= 0 and for S-shaped bifurcation curves withdifferent asymptotic behaviors for the bifurcation curves when0 <= p <= 1. In particular, when p = 0 and 1, our resultsimprove those of Hastings and McLeod [Proc. Roy. Soc. EdinburghSect.A 101(1985),15-30 ]. While our results for 0<p<1generalize that of Hasting and McLeod and of Dancer [J.Differential Equations 37 (1980),404-437] in which p is assumedto be a nonnegative integer.