Critical quasilinear Schroedinger equations with electromagnetic fields

Abstract

The p-Laplace operator in the entire N-dimensional Euclidean space, subject to external electromagnetic potentials, is investigated. In the general case 1<p<N, the existence of at least one solution of mountain pass type to a weighted critical equation is proved. Our technique relies on variational methods and faces a twofold difficulty: double lack of compactness, which requires concentration compactness arguments; and a complex quasilinear framework, which entails appropriate inequalities.