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Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem. (English) Zbl 1196.35109
Summary: Let $𝛺$ be a ${C}^{1,1}$-bounded domain in ${ℝ}^{n}$ for $n⩾2$. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem ${\Delta }u+a\left(x\right){u}^{-\sigma }=0$ in $𝛺$, ${u}_{|\partial 𝛺}=0$, where $\sigma ⩾0$, $a$ is a nonnegative function in ${C}_{\text{loc}}^{\alpha }\left(𝛺\right)$, $0<\alpha <1$ and there exists $c>0$ such that $\frac{1}{c}⩽a\left(x\right){\left(\delta \left(x\right)\right)}^{\lambda }{\prod }_{k=1}^{m}{\left({\text{Log}}_{k}\left(\frac{\omega }{\delta \left(x\right)}\right)\right)}^{{\mu }_{k}}⩽c$. Here, $\lambda ⩽2,{\mu }_{k}\in ℝ$, $\omega$ is a positive constant and $\delta \left(x\right)=\text{dist}\left(x,\partial 𝛺\right)$.
##### MSC:
 35J75 Singular elliptic equations 35J25 Second order elliptic equations, boundary value problems 35B09 Positive solutions of PDE 35B40 Asymptotic behavior of solutions of PDE
##### Keywords:
asymptotic behavior; Dirichlet problem; singular equation
##### References:
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