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Singular solutions of a nonlinear equation in a punctured domain of <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^{2}$\end{document}</tex-math></inline-formula>
We consider the following singular semilinear problem \begin{document}$\left\{ \begin{array}{l} - \Delta u(x) = a(x){u^\sigma }(x),{\rm{ }}x \in \Omega \backslash \{ 0\} ({\rm{in\;the\;distributional\;sense}}),\\\;u > 0,{\rm{ on}}\;\Omega \backslash \{ 0\} ,\\\mathop {\lim }\limits_{\left| x \right| \to 0} \frac{{u(x)}}{{\ln \left| x \right|}} = 0,\\u(x) = 0,\;x \in \partial \Omega ,\end{array} \right.$ \end{document} where …