Boundary layer and variational eigencurve in two-parameter single pendulum type equations

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We consider the nonlinear two-parameter single pendulum type equation $-u''(t) + \mu f(u(t)) = \lambda\sin u(t), t \in I$ :$= (-T, T), u(t) > 0, t \in I, u(\pm T) = 0$, where $T > 0$ is a constant and $\mu, \lambda > 0$ are parameters. For a given $\mu > 0$, there exists a solution triple $(\mu, \lambda(\mu), u_\mu) \in \mbox{\bf R}_+^2 \times C^2(\bar{I})$, which is obtained by a variational method, such that $u_\mu$ develops a boundary layer as $\mu \to \infty$. We establish the precise asymptotic formulas for $||u_\mu||_\infty, u_\mu'(\pm T)$ and the variational eigencurve $\lambda(\mu)$ as $\mu \to \infty$.