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Annali di Matematica (2016) 195:1737–1748DOI 10.1007/s10231-015-0545-4Existence of solution for a general class of ellipticequations with exponential growthAnderson L. A. de Araujo 1 · Marcelo Montenegro 2Received: 7 October 2014 / Accepted: 26 November 2015 / Published online: 12 December 2015© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2015Abstract We use Galerkin approximations to show the existence of solution for a class ofelliptic equations on bounded domains in R 2 with subcritical or critical exponential nonlin-earities. We are able to solve the problem under more general assumptions usually assumedin the variational the approach, but not in our paper.Keywords Dirichlet problem · Galerkin approximation · Trudinger–Moser inequality ·Exponential growth · Conformal geometryMathematics Subject Classif i cation 35B38 · 35J92 · 35B33 · 35J621 IntroductionWe prove the existence of solution of the problem???−?v = λv q + f (v) in ?v > 0 in ?v = 0 on ∂?,(1)This author was partially supported by FAPESP, Brazil, Grant 2013/22328-8. This author was partiallysupported by CNPq, Brazil.B Anderson L. A. de Araujoanderson.araujo@ufv.brMarcelo Montenegromsm@ime.unicamp.br1UFV, Viçosa, Brazil2Universidade Estadual de Campinas, Campinas, Brazil123