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

Abstract: 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 domain in $\mathbb{R}^{N}$ with $N\geq2$, $b\in \rm C^{\infty}(\Omega)$ is positive in $\Omega$ and may be singular or vanish on the boundary, $f\in C^{\infty}(0,\infty)\cap C[0, +\infty)$ (or $f\in C^{\infty}(\mathbb{R})$) is positive and increasing on $[0, +\infty)$ (or $\mathbb{R}$) and satisfies the Keller-Osserman type condition. We first supply an upper and lower solution method of classical $k$-convex large solutions to the above equation, and then we studied the optimal global estimate and boundary behavior of large solutions. In particular, we investigate the asymptotic behavior of such solutions when the parameters on $b$ tend to the corresponding critical values and infinity.