標題: Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions
作者: Liang, Yu-Hao
Wang, Shin-Hwa
應用數學系
Department of Applied Mathematics
公開日期: 15-Jun-2016
摘要: We study the classification and evolution of bifurcation curves of positive solutions for the onedimensional perturbed Gelfand equation with mixed boundary conditions given by {u"(x) + lambda exp(au/a+u) = 0, 0 < x < 1, u(0) = 0, u\' (1) = -c < 0. We prove that, for positive a <= a(0) (approximate to 0.501) and c > 0, the bifurcation curve is strictly increasing on the (A, II ulloo)-plane, and there exists a positive Xo such that the problem has no positive solution for 0 < lambda < lambda(0) and exactly one positive solution for lambda > lambda(0). While for a >= a(1) (approximate to 4.107), there exists c(1) (= c(1) (a)) > 1.057 such that, on the (A, 11u1100) -plane, (i) when 0 < c < cl, the bifurcation curve is S-shaped, and the problem has at least three positive solutions for some range of positive A; (ii) when c > cl, the bifurcation curve is c-shaped and the problem has at least two positive solutions for some range of positive A. (C) 2016 Elsevier Inc. All rights reserved.
URI: http://dx.doi.org/10.1016/j.jde.2016.02.021
http://hdl.handle.net/11536/133611
ISSN: 0022-0396
DOI: 10.1016/j.jde.2016.02.021
期刊: JOURNAL OF DIFFERENTIAL EQUATIONS
Volume: 260
Issue: 12
起始頁: 8358
結束頁: 8387
Appears in Collections:Articles