Bifurcation from S-shaped solution curves in a class of Sturm-Liouville problems related to climate modeling (Q2733864)

The author deals with a class of parameter-dependent Legendre-type ordinary differential equations of the form \(-(kpu')'=\mu f(u)-g(u)\), which arise from energy-balance climate models; \(k\) is a positive function on \([-1,1]\), \(p(x)=1-x^2\); \(f\) and \(g\) are nonnegative functions on \([0,\infty\)) and \(\mu\) is a nonnegative parameter. Under suitable conditions on \(f\) and \(g\) it is shown that the set of all pairs \((\mu ,u)\), where \(\mu\) is in \([0,\infty)\) and \(u\) is a classical solution to the differential equation, contains an ``\(S\)-shaped'' branch of trivial solutions, i.e. pairs \((\mu ,u)\) with \(u= \text{const}\). The author obtains a sufficient condition for the bifurcation of nontrivial solutions from the trivial branch and describes the global behavior of the bifurcating branches.