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

  • * Corresponding author: Imed Bachar

    * Corresponding author: Imed Bachar 
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  • 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 $0$.

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


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