论文标题
大型解决方案的最佳全球估计和边界行为
The optimal global estimate and boundary behavior for large solutions to the k-Hessian equation
论文作者
论文摘要
在本文中，我们考虑$ k $ -Hessian方程$ s_ {k}（d^{2} u）= b（x）f（x）f（u）\ mbox {in}ω，\，\，u =++\ infty \ mbox {on} \partialΩ$ $ \ mathbb {r}^{n} $带有$ n \ geq2 $，$ b \ in \ rm c^{\ infty}（ω）$在$ω$中是正面的，并且在边界上可能是单数或消失的，$ f \ in c^{\ infty}（\ infty}（0，\ infty} $ f ty， c^{\ infty}（\ mathbb {r}）$是正面的，并且在$ [0， +\ infty）$（或$ \ mathbb {r} $）上增加，并满足Keller-osserman类型条件。我们首先为上述方程式提供了经典$ K $ -CONVEX大解决方案的上层和下溶液方法，然后我们研究了大解决方案的最佳全球估计和边界行为。特别是，当$ b $上的参数倾向于相应的临界值和无穷大时，我们研究了这种解决方案的渐近行为。
In this paper, we consider the $k$-Hessian equation $S_{k}(D^{2}u)=b(x)f(u)\mbox{ in }Ω,\,u=+\infty \mbox{ on }\partialΩ$, where $Ω$ is a smooth, bounded, strictly convex domain in $\mathbb{R}^{N}$ with $N\geq2$, $b\in \rm C^{\infty}(Ω)$ is positive in $Ω$ 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.
