On an indefinite nonhomogeneous equation with critical exponential growth

In this manuscript it is obtained existence of solution for the equation -div(a(|del u|p)|del u|p-2 del u)+b(|u|p)|u|p-2u=c(x)f(u),inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\text {div}(a(|\nabla u|<^>p)|\nabla u|<^>{p-2}\nabla u) +b(|u|<^>p)|u|<^>{p-2}u = c(x)f(u), \ \text {in} \ {\mathbb {R}}<^>N, \end{aligned}$$\end{document}where 1<p<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<N$$\end{document}, N >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}, the functions a,b:R+-> R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b:{\mathbb {R}}<^>+\rightarrow {\mathbb {R}}<^>+$$\end{document} satisfy suitable conditions, c is a continuous sign-changing potential and the nonlinearity f has an exponential critical growth at infinity. In the proof we apply variational methods.