A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem

作者：Hung Kuo Chih; Wang Shin Hwa^*

来源：Journal of Differential Equations, 2011, 251(2): 223-237.

DOI：10.1016/j.jde.2011.03.017

• Positive solution
• Exact multiplicity
• S-shaped bifurcation curve
• Positone problem
• Convex-concave nonlinearity
• Perturbed Gelfand problem

We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem
{u ''(x) + lambda f (u) = 0, -1 < x < 1,
u(-1) = u(1) = 0,
where lambda > 0 is a bifurcation parameter, f is an element of C(2)[0, infinity) satisfies f(0) > 0 and f (u) > 0 for u > 0, and f is convex-concave on (0, proportional to). Under a mild condition, we prove that the bifurcation curve is S-shaped on the (lambda, parallel to u parallel to(infinity))-plane. We give an application to the perturbed Gelfand problem
{u ''(x) +lambda exp(au/a+u) = 0, -1 < x < 1,
u(-1) = u(1) = 0,
where a > 0 is the activation energy parameter. We prove that, if a >= a*approximate to 4.166, the bifurcation curve is S-shaped on the (lambda, parallel to u parallel to(infinity))-plane. Our results improve those in [S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475-1485] and P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999) 1011-1020.

• 出版日期2011-7-15
• 单位清华大学