A log–exp elliptic equation in the plane

<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in id="M2">\begin{document}$ \Omega id="M3">\begin{document}$ 0 on id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where id="M5">\begin{document}$ is smooth bounded domain id="M6">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula>. We replace function id="M7">\begin{document}$ \log(u) by id="M8">\begin{document}$ g_\epsilon(u) which pointwisely converges to -<inline-formula><tex-math id="M9">\begin{document}$ as id="M10">\begin{document}$ \epsilon \rightarrow When parameter id="M11">\begin{document}$ \lambda>0 small enough, corresponding energy functional perturbed equation id="M12">\begin{document}$ has critical point id="M13">\begin{document}$ u_\epsilon id="M14">\begin{document}$ H_0^1(\Omega) nontrivial original id="M15">\begin{document}$ $\end{document}</tex-math></inline-formula>.</p>