Existence of positive solutions to anisotropic elliptic equations in the entire Euclidean space

Abstract

In this paper, we research an anisotropic elliptic equation in the entire Euclidean space R^{N} . Under the assumption that the nonlinearity of the equation has exponential or critical growth, we prove, by means of the mountain-pass theorem, that the equation admits a nontrivial positive weak solution. Moreover, applying the mountain-pass theorem and Ekeland's variational principle, we show that a perturbation of the equation possesses two distinct positive weak solutions.