We consider the nonlinear two-parameter problem, which comes from a perturbed simple pendulum problem \documentclass[12pt]{minimal}
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$$-u^{\prime\prime}(t)+\mu f(u(t)) = \lambda g(u(t)), t \in I: = (-T, T), u(t) > 0,\quad t \in I,\quad u(\pm T) = 0,$$
\end{document} where μ, λ > 0 are parameters and T > 0 is a constant. For a given μ > 0, there exists a solution triple (\documentclass[12pt]{minimal}
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$$\mu, \lambda(\mu), u_{\mu}) \in \rm{R}^{2}_{+}\times C^{2}(\bar{I}),$$
\end{document} which is obtained by a variational method, such that uμ is almost flat inside I and develops boundary layers as \documentclass[12pt]{minimal}
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$$\mu \rightarrow \infty$$
\end{document}. We establish the precise asymptotic formulas for \documentclass[12pt]{minimal}
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$$\|u_{\mu}\|_{q}(1 \leq q \leq \infty), u^{\prime}_{\mu} (\pm T)$$
\end{document} and the variational eigencurve λ(μ) as \documentclass[12pt]{minimal}
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$$\mu \rightarrow \infty$$
\end{document}. By these formulas, we understand well not only the local behavior of uμ as \documentclass[12pt]{minimal}
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$$\mu \rightarrow \infty$$
\end{document}, but also the total shape of uμ. Furthermore, we find the precise asymptotics of \documentclass[12pt]{minimal}
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$$\|u_{\mu}\|_{q}$$
\end{document} as \documentclass[12pt]{minimal}
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$$q \rightarrow \infty$$
\end{document}. By this, we understand well how \documentclass[12pt]{minimal}
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$$\|u_{\mu}\|_{q}$$
\end{document} tends to \documentclass[12pt]{minimal}
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$$\|u_{\mu}\|_{\infty}$$
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$$q \rightarrow \infty$$
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