GLOBAL BEHAVIOR OF BIFURCATION CURVES FOR THE NONLINEAR EIGENVALUE PROBLEMS WITH PERIODIC NONLINEAR TERMS

We consider the bifurcation problem -u ''(t) = lambda(u(t) + g(u(t))), u(t) > 0, t is an element of I : = (-1, 1), u(+/- 1) = 0, where g(u) is an element of C-1(R) is a periodic function with period 2 pi and lambda > 0 is a bifurcation parameter. It is known that, under the appropriate conditions on g, lambda is parameterized by the maximum norm alpha = parallel to u(lambda)parallel to(infinity) of the solution u(lambda) associated with lambda and is written as lambda = lambda(alpha). If g(u) is periodic, then it is natural to expect that lambda(alpha) is also oscillatory for alpha >> 1. We give a simple condition of g(u), by which we can easily check whether lambda(alpha) is oscillatory and intersects the line lambda = pi(2)/4 infinitely many times for alpha >> 1 or not.