Exact multiplicity of solutions and S-shaped bifurcation curves for the p-Laplacian perturbed Gelfand problem in one space variable

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:342

Article

Wang, Shin-Hwa¹  Yeh, Tzung-Shin¹ 

[1] Natl Tsing Hua Univ, Dept Math, Hsinchu 300, Taiwan

关键词: exact multiplicity;    positive solution;    S-shaped bifurcation curve;    p-Laplacian;    perturbed Gelfand problem;    time map;   

DOI  :  10.1016/j.jmaa.2007.12.026

来源: Elsevier

【 摘 要 】

We study exact multiplicity of positive solutions and the bifurcation curve of the p-Laplacian perturbed Gelfand problem from combustion theory {(phi(p)(u'(x)))' + lambda exp(au/a + u) = 0, -1 < x < 1, u(-1) = u(1) = 0, where p > 1, phi(p)(y) = vertical bar y vertical bar(p-2)y, (phi(p)(u'))' is the one-dimensional p-Laplacian, lambda > 0 is the Frank-Kamenetskii parameter, u(x) is the dimensionless temperature, and the reaction term f (u) = exp(au/a+u) is the temperature dependence obeying the Arrhenius reaction-rate law. We find explicitly (a) over tilde = (a) over tilde (p) > 0 such that, if the activation energy a >= (a) over tilde, then the bifurcation curve is S-shaped in the (lambda, parallel to u parallel to(infinity))-plane. More precisely, there exist 0 < lambda(*) < lambda* < infinity such that the problem has exactly three positive solutions for lambda(*) < lambda < lambda*, exactly two positive solutions for lambda = lambda(*) and lambda = lambda*, and a unique positive solution for 0 < lambda < lambda(*) and lambda* < lambda < infinity. (c) 2007 Elsevier Inc. All rights reserved.

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