| Summary: | We consider the nonlinear eigenvalue problem
$$\displaylines{
u''(t) + \lambda (u(t)^p - u(t)^q) = 0, \quad u(t) > 0,\quad -1<t<1,\cr
u(1) = u(-1) = 0,
}$$
where $1 < p < q$ are constants and $\lambda > 0$ is a parameter.
It is known in [13] that the bifurcation curve $\lambda(\alpha)$
consists of two branches, which are denoted by $\lambda_\pm(\alpha)$.
Here, $\alpha = \| u_\lambda\|_\infty$. We establish the
asymptotic behavior of the turning point $\alpha_p$
of $\lambda(\alpha)$, namely, the point which satisfies
$d\lambda(\alpha_p)/d\alpha = 0$ as $p \to q$ and $p \to 1$.
We also establish the asymptotic formulas for $\lambda_{+}(\alpha)$
and $\lambda_{-}(\alpha)$ as $\alpha \to 1$ and $\alpha \to 0$, respectively.
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