Quasilinear asymptotically periodic Schrödinger equations with critical growth

Abstract

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in \({\mathbb{R}^N}\) with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in \({H^1(\mathbb{R}^N)}\) and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.