ASYMPTOTIC FORMULAS FOR OSCILLATORY BIFURCATION DIAGRAMS OF SEMILINEAR ORDINARY DIFFERENTIAL EQUATIONS

We study the nonlinear eigenvalue problem -u ''(t) = lambda (u(t)(p) + g(u(t)))), u(t) > 0, t is an element of (-1, 1), u(+/- 1) = 0 where g(u) = h(u) sin(u(r)), p, r are given constants satisfying p >= 0, 0 < r <= 1 and lambda > 0 is a parameter. It is known that under suitable conditions on h, lambda is parameterized by the maximum norm alpha = parallel to u(alpha)parallel to(infinity) of the solution u(lambda) associated with lambda and lambda = lambda(alpha) is a continuous function for alpha > 0. When p = 1, h(u) 1 and r = 1/2, this equation has been introduced by Chen [4] as a model equation such that there exist infinitely many solutions near lambda = pi(2)/4. We prove that lambda(alpha) is an oscillatory bifurcation curve as alpha -> infinity by showing the asymptotic formula for lambda(alpha). It is found that the shapes of bifurcation curves depend on the condition p > 1 or p < 1. The main tools of the proof are time-map argument and stationary phase method.