Existence of solution for a general class of elliptic equations with exponential growth

We use Galerkin approximations to show the existence of solution for a class of elliptic equations on bounded domains in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} with subcritical or critical exponential nonlinearities. We are able to solve the problem under more general assumptions usually assumed in the variational the approach, but not in our paper.