Critical quasilinear Schroedinger equations with electromagnetic fields

TL;DR

This paper investigates the existence of solutions to a weighted critical p-Laplace equation with electromagnetic potentials in Euclidean space, employing variational methods to overcome compactness and quasilinear challenges.

Contribution

It establishes the existence of mountain pass solutions for weighted critical quasilinear equations with electromagnetic fields using advanced variational techniques.

Findings

Existence of solutions proved for 1<p<N

Overcomes double lack of compactness with concentration compactness

Develops inequalities suitable for complex quasilinear framework

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.