Multiplicity results for generalized quasilinear critical Schr\"odinger equations in R^N

TL;DR

This paper establishes multiple solutions, including positive and negative energy solutions, for a generalized quasilinear Schrödinger equation with critical and subcritical nonlinearities in R^N, using advanced variational methods.

Contribution

It provides new multiplicity and nonexistence results for a class of generalized quasilinear Schrödinger equations involving critical and subcritical nonlinearities with weights.

Findings

Proved existence of multiple solutions with positive and negative energy.

Established nonexistence results under certain conditions.

Applied variational methods and concentration compactness principles.

Abstract

Multiplicity results are proved for solutions both with positive and negative energy, as well as nonexistence results, of a generalized quasilinear Schr\"odinger potential free equation in the entire R^N involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation has been derived from models of several physical phenomena such as superfluid film in plasma physics as well as the self-channelling of a high-power ultra-short laser in matter. Proof techniques, also in the symmetric setting, are based on variational tools, including concentration compactness principles, to overcome lack of compactness, and the use of a change of variable in order to deal with a well defined functional.