Existence of positive solutions to anisotropic elliptic equations in bounded domain of <inline-formula><tex-math id="M1">$ \mathbb{R}^{N} $</tex-math></inline-formula>

In this paper, an anisotropic elliptic equation in a bounded domain of $ \mathbb{R}^{N} $ is considered. Assume that the nonlinearity of the equation has exponential growth or critical growth. It is proved that this equation has a nontrivial positive weak solution by the mountain-pass theorem. Furthermore, it is shown that there are two distinct positive solutions to a perturbation of the equation by applying the anisotropic Moser-Trudinger inequality, mountain-pass theorem, and Ekeland's variational principle.