A theorem on reversed S-shaped bifurcation curves for a class of boundary value problems and its application

作者：Wang Shin Hwa; Yeh Tzung Shin^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2009, 71(1-2): 126-140.

DOI：10.1016/j.na.2008.10.070

• S-shaped bifurcation curve
• Positive solution
• Exact multiplicity
• Time map

We study the bifurcation curves of positive solutions of the boundary value problem
{u ''(x) + f(epsilon)(u(x)) = 0, -1 < x < 1, u(-1) = u(1) = 0,
where f(epsilon)(u) = g(u) - epsilon h(u), epsilon is an element of R is a bifurcation parameter, and functions g, h is an element of C[0, infinity) boolean AND C(2)(0, infinity) satisfy five hypotheses presented herein. Assuming these hypotheses on fixed g and h, we prove that the bifurcation curve is reverse S-shaped on the (epsilon, parallel to u parallel to(infinity))-plane; that is, the bifurcation curve has exactly two turning points at some points ((epsilon) over tilde, parallel to u((epsilon) over tilde)parallel to(infinity)) and (epsilon*, parallel to u(epsilon*)parallel to(infinity)) such that (epsilon) over tilde < epsilon* and parallel to u(<(epsilon)over tilde>)parallel to(infinity) < parallel to u(epsilon*)parallel to(infinity). In addition, we prove that epsilon* > 0. Thus the exact number of positive solutions can be precisely determined by the values (epsilon) over tilde. and epsilon*. We give an application to the two-parameter bifurcation problem
{u ''(x) + lambda(1 + u(2) - epsilon u(3)) = 0, -1 < x < 1, u(-1) = u(1) = 0,
where lambda, epsilon are two positive bifurcation parameters. Some new results are obtained.

• 出版日期2009-7-1
• 单位清华大学