We study the nonlinear two-parameter problem $-u^{\prime\prime}(x) + \lambda u(x)^q = \mu u(x)^p$ , $u(x) > 0$ , $ x \in (0, 1)$ , $u(0) = u(1) = 0$ . Here, $1 #60; q #60; p$ are constants and $\lambda,\mu > 0$ are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve $\mu(\lambda)$ as $\lambda \rightarrow \infty$ . We emphasize that the critical case concerning the decaying rate of the second term is $p = (3q-1)/2$ and this kind of criticality is new for two-parameter problems.