GLOBAL AND LOCAL BEHAVIOR OF THE BIFURCATION DIAGRAMS FOR SEMILINEAR PROBLEMS

We consider the nonlinear eigenvalue problem u"(t) + lambda(u(t)(p) u(t)(q)) = 0, u(t) > 0, -1 < t < 1, u(1) = u(-1) = 0, where 1 < p < q are constants and lambda > 0 is a parameter. It is known in [13] that the bifurcation curve lambda(alpha) consists of two branches, which are denoted by lambda +/-(alpha). Here, alpha = vertical bar vertical bar u(lambda)vertical bar vertical bar(infinity). We establish the asymptotic behavior of the turning point alpha(p), of lambda(alpha), namely, the point which satisfies d lambda(alpha(p))/d alpha = 0 as p -> q and p -> 1. We also establish the asymptotic formulas for lambda+(alpha) and lambda-(alpha) as alpha -> 1 and alpha -> 0, respectively.