Asymptotic Shape of Solutions to the Perturbed Simple Pendulum Problems

We consider the positive solution of the perturbed simple pendulum problem u′′(r) + N − 1 r u′(r)− g(u(t)) + λ sinu(r) = 0, with 0 < r < R, u′(0) = u(R) = 0. To understand well the shape of the solution uλ when λ 1, we establish the leading and second terms of ‖uλ‖q (1 ≤ q < ∞) with the estimate of third term as λ → ∞. We also obtain the asymptotic formula for uλ(R) as λ→∞.