We study the asymptotic behavior, as \(\gamma \) tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is

$$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$

where \(\Omega \) is an open, bounded subset of \({\mathbb {R}}^{N}\) and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of \(\Omega \) or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with

$$\begin{aligned} -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$

中文翻译：

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奇异椭圆方程的渐近行为和解的存在性

我们研究与奇异半线性椭圆型方程有关的齐次Dirichlet问题的解的渐近行为，因为\（\ gamma \）趋于无穷大，其模型为

$$ \ begin {aligned}-\ Delta u = \ frac {f（x）} {u ^ \ gamma} \，\ text {in} \ Omega，\ end {aligned} $$

其中\（\欧米茄\）是一个开放的，有界的子集\（{\ mathbb {R}} ^ {N} \）和˚F是有界函数。我们在f的两个不同假设下处理极限方程的存在：在\（\ Omega \）的每个紧凑包含子集上严格为正，或者仅为非负。通过这项研究，我们推导了与奇异Dirichlet问题相关的正解的最优存在结果。

$$ \ begin {aligned}-\ Delta v + \ frac {| \ nabla v | ^ 2} {v} = f \，\ text {in} \ Omega。\ end {aligned} $$