Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in R^n

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of weak solutions for nonhomogeneous quasilinear elliptic equations of the formwhere n ≥ 2, ∆nu ≡ div(|∇u| n−2 ∇u) is the n-Laplacian and ε is a positive parameter. Here the function g(x) may be unbounded in x and the nonlinearity f (s) has critical growth in the sense of Trudinger-Moser inequality, more precisely f (s) behaves like e α 0 |s| n/(n−1) when s → +∞ for some α 0 > 0. Under some suitable assumptions and based on a Trudinger-Moser type inequality, our results are proved by using Ekeland variational principle, minimization and mountain-pass theorem.