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In this paper, we consider the $k$-Hessian equation$S_{k}(D^{2}u)=b(x)f(u)\mbox{ in }\Omega,\,u=+\infty \mbox{ on}\partial\Omega$, where $\Omega$ is a smooth, bounded, strictly convex domainin $\mathbb{R}^{N}$ with $N\geq2$, $b\in \rm C^{\infty}(\Omega)$ is positive in$\Omega$ and may be singular or vanish on the boundary, $f\inC^{\infty}(0,\infty)\cap C[0, +\infty)$ (or $f\in C^{\infty}(\mathbb{R})$) ispositive and increasing on $[0, +\infty)$ (or $\mathbb{R}$) and satisfies theKeller-Osserman type condition. We first supply an upper and lower solutionmethod of classical $k$-convex large solutions to the above equation, and thenwe studied the optimal global estimate and boundary behavior of largesolutions. In particular, we investigate the asymptotic behavior of suchsolutions when the parameters on $b$ tend to the corresponding critical valuesand infinity.

The optimal global estimate and boundary behavior for large solutions to the k-Hessian equation