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

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• 作者：Shin-Hwa Wang ; Tzung-Shin Yeh
• 关键词：Exact multiplicity ; Positive solution ; S-shaped bifurcation curve ; p-Laplacian ; Perturbed Gelfand problem ; Time map
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2008
• 出版时间：15 June 2008
• 年：2008
• 卷：342
• 期：2
• 页码：1175-1191
• 全文大小：239 K

文摘

We study exact multiplicity of positive solutions and the bifurcation curve of the p-Laplacian perturbed Gelfand problem from combustion theory

where p>1, φ_p(y)=|y|^p−2y, (φ_p(u^′))^′ is the one-dimensional p-Laplacian, λ>0 is the Frank–Kamenetskii parameter, u(x) is the dimensionless temperature, and the reaction term is the temperature dependence obeying the Arrhenius reaction-rate law. We find explicitly such that, if the activation energy , then the bifurcation curve is S-shaped in the (λ,u_∞)-plane. More precisely, there exist 0<λ_*<λ^*<∞ such that the problem has exactly three positive solutions for λ_*<λ<λ^*, exactly two positive solutions for λ=λ_* and λ=λ^*, and a unique positive solution for 0<λ<λ_* and λ^*<λ<∞.