This paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models $-△u\pm \lambda \frac{{\left|\nabla u\right|}^2}{u^{\beta }}=b\left(x\right)u^{-\alpha }$, $u>0$, $x\in \Omega$, ${u|}_{\partial \Omega }=0$, where Ω is a bounded domain with smooth boundary in ${ℝ}^N$, $\lambda >0$, $\beta >0$, $\alpha >-1$, and $b\in C_{\mathrm{loc}}^{\nu }\left(\Omega \right)$ for some $\nu \in (0,1)$, and b is positive in Ω but may be vanishing or singular on $\partial \Omega$. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

中文翻译：

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具有自然增长的两类非线性奇异Dirichlet问题：存在性和渐近行为

本文关注两类模型经典解的存在性，唯一性和渐近行为 $--△ü\pm \lambda \frac{{\left|\nabla ü\right|}^2}{{ü}^{\beta }}=b（X）{ü}^{--\alpha }$， $ü>0$， $X\in \Omega$， ${ü|}_{\partial \Omega }=0$，其中Ω是在 ${ℝ}^{ñ}$， $\lambda >0$， $\beta >0$， $\alpha >--\mathrm{1个}$和 $b\in C_{\mathrm{位置}}^{\nu }（\Omega ）$ 对于一些 $\nu \in （0，\mathrm{1个}）$，并且b的正值为Ω，但在b上可能消失或为奇数$\partial \Omega$。我们的方法主要基于非线性变换以及合适的子解和超解的构造。