A log–exp elliptic equation in the plane

In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely \begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document} in \begin{document}$ \Omega $\end{document} with \begin{document}$ u = 0 $\end{document} on \begin{document}$ \partial\Omega $\end{document} , where \begin{document}$ \Omega $\end{document} is a smooth bounded domain in \begin{document}$ \mathbb{R}^{2} $\end{document} . We replace the singular function \begin{document}$ \log(u) $\end{document} by a function \begin{document}$ g_\epsilon(u) $\end{document} which pointwisely converges to - \begin{document}$ \log(u) $\end{document} as \begin{document}$ \epsilon \rightarrow 0 $\end{document} . When the parameter \begin{document}$ \lambda>0 $\end{document} is small enough, the corresponding energy functional to the perturbed equation \begin{document}$ -\Delta u + g_\epsilon(u) = \lambda f(u) $\end{document} has a critical point \begin{document}$ u_\epsilon $\end{document} in \begin{document}$ H_0^1(\Omega) $\end{document} , which converges to a nontrivial nonnegative solution of the original problem as \begin{document}$ \epsilon \rightarrow 0 $\end{document} .