Three-term spectral asymptotics for nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem¶¶ $ -u''(t) + \vert u(t)\vert^{p-1}u(t) + f(u(t)) = \lambda u(t),\\ \enskip t \in I := (0, 1), \enskip u(0) = u(1) = 0, $ ¶¶ where $ p > 1 $ is a constant and $ \lambda > 0 $ is an eigenvalue parameter. We establish the three-term asymptotics of $ n-th $ eigencurve $ \lambda_n(\alpha) $ (associated with eigenfunction $ u_{n,\alpha} $ with $ n-1 $ simple interior zeros and $ \Vert u_{n,\alpha} \Vert_2 = \alpha $ ) as $ \alpha \to \infty $ . We also obtain the corresponding asymptotics of the width of the boundary layer of $ u_{n,\alpha} $ as $ \alpha \to \infty $ .