We study the asymptotic behavior, as γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is -Δu=f(x)uγinΩ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is an open, bounded subset of RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{N}$$\end{document} 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 Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with -Δv+|∇v|2v=finΩ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$\end{document}
