ASYMPTOTIC SHAPE OF SOLUTIONS TO THE PERTURBED SIMPLE PENDULUM PROBLEMS

We consider the positive solution of the perturbed simple pendulum problem u ''(r) + N - 1r u'(r) - g(u(t)) + lambda sin u(r) = 0, with 0 < r < R, u'(0) = u(R) = 0. To understand well the shape of the solution u lambda when lambda >> 1, we establish the leading and second terms of parallel to u lambda parallel to(q) (1 <= q < infinity) with the estimate of third term as lambda -> infinity. We also obtain the asymptotic formula for u'(lambda) (R) as lambda -> infinity.