A GLOBAL BIFURCATION THEOREM FOR A POSITONE MULTIPARAMETER PROBLEM AND ITS APPLICATION

作者：Hung Kuo Chih; Huang Shao Yuan^*; Wang Shin Hwa

来源：Discrete and Continuous Dynamical Systems, 2017, 37(10): 5127-5149.

DOI：10.3934/dcds.2017222

• Global bifurcation
• multiparameter problem
• S-shaped bifurcation curve
• exact multiplicity
• positive solution

We study the global bifurcation and exact multiplicity of positive solutions for the positone multiparameter problem
{u"(x) + lambda f(epsilon)(u) = 0, -1 < x < 1, u(-1) = u(1) = 0,
where lambda > 0 is a bifurcation parameter and epsilon > 0 is an evolution parameter. Under some suitable hypotheses on f(epsilon)(u), we prove that there exists (epsilon) over tilde > 0 such that, on the (lambda, vertical bar vertical bar u vertical bar vertical bar(infinity))-plane, the bifurcation curve is S-shaped for 0 < epsilon < (epsilon) over tilde and is monotone increasing for epsilon >= (epsilon) over tilde. We give an application for this problem with a class of polynomial nonlinearities f(epsilon)(u) = -epsilon u(P) + bu(2) + cu + d of degree p >= 3 and coefficients epsilon, b, d > 0, c >= 0. Our results generalize those in Hung and Wang

• 出版日期2017-10
• 单位清华大学