Singular solutions of a nonlinear equation in apunctured domain of $\mathbb{R}^{2}$

Imed Bachar; Habib Mâagli

doi:10.3934/dcdss.2019012

Article Contents

2019, Volume 12, Issue 2: 171-188. Doi: 10.3934/dcdss.2019012

This issue Previous Article Nonlinear equations involving the square root of the Laplacian Next Article On solutions of semilinear upper diagonal infinite systems of differential equations

Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$

Imed Bachar^1, , and
Habib Mâagli^2,

1.
King Saud University, College of Science, Mathematics Department, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
King Abdulaziz University, College of Sciences and Arts, Rabigh Campus, Department of Mathematics, P.O. Box 344, Rabigh 21911, Saudi Arabia

* Corresponding author: Imed Bachar
* Corresponding author: Imed Bachar

Received: April 30, 2017

Revised: November 30, 2017

Published: April 2019

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Abstract

We consider the following singular semilinear problem

$\left\{ \begin{array}{l} - \Delta u(x) = a(x){u^\sigma }(x),{\rm{ }}x \in \Omega \backslash \{ 0\} ({\rm{in\;the\;distributional\;sense}}),\\\;u > 0,{\rm{ on}}\;\Omega \backslash \{ 0\} ,\\\mathop {\lim }\limits_{\left| x \right| \to 0} \frac{{u(x)}}{{\ln \left| x \right|}} = 0,\\u(x) = 0,\;x \in \partial \Omega ,\end{array} \right.$

where $σ <1,$ $Ω $ is a bounded regular domain in $\mathbb{R}^{2}$ with $0∈ Ω .$ The weight function $a(x)$ is requiredto be positive and continuous in $Ω \backslash \{0\}$ with thepossibility to be singular at $x = 0$ and/or at the boundary $\partial Ω. $ When the function $a$ satisfies sharp estimates related to Karamataclass, we prove the existence and global asymptotic behavior of a positivecontinuous solution on $\overline{Ω }\backslash \{0\}$ which couldblow-up at $0$.

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Mathematics Subject Classification: Primary: 34B16, 34B27; Secondary: 34B18.

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