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# Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

Zhijun Zhang
• Corresponding author
• School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, P. R. China
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Published Online: 2019-08-27 | DOI: https://doi.org/10.1515/ans-2019-2054

## Abstract

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

MSC 2010: 35J25; 35J65; 35J67

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Revised: 2019-06-24

Accepted: 2019-06-26

Published Online: 2019-08-27

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571295

This work is supported in part by NSF of P. R. China under grant 11571295.

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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