| Abstract: |
| We are concerned with the boundary blowup problem for the so-called $k$-Hessian equation of the form $F_k[u] = f(x)g(u)$ in a uniformly $(k-1)$-convex bounded domain $\Omega \subset \mathbb{R}^n$, where $f(x)$ behaves like $\text{dist}(x,\partial\Omega)^{\alpha}$ as $\text{dist}(x,\partial\Omega) \to 0$ and $g(u)$ behaves like $u^p$ as $u \to \infty$. We establish the blowup rate of a solution near the boundary $\partial \Omega$. Also, we consider the boundary blowup problem for the so-called $k$-curvature equation of the form $H_k[u] = g(u) h(|Du|)$ in a uniformly $k$-convex (or uniformly convex) bounded domain $\Omega \subset \mathbb{R}^n$, where $g(u)$ behaves like $u^p$ as $u \to \infty$ and $h(s)$ behaves like $s^{-q}$ as $s \to \infty$, and establish the blowup rate of a solution near $\partial \Omega$. |
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