  In this paper a quasi-linear elliptic equation in the whole Euclidean space
is considered. The nonlinearity of the equation is assumed to have exponential
growth or have critical growth in view of Trudinger-Moser type inequality.
Under some assumptions on the potential and the nonlinearity, it is proved that
there is a nontrivial positive weak solution to this equation. Also it is shown
that there are two distinct positive weak solutions to a perturbation of the
equation. The method of proving these results is combining Trudinger-Moser type
inequality, Mountain-pass theorem and Ekeland's variational principle.
