Local and nonlocal critical growth anisotropic quasilinear elliptic systems

TL;DR

This paper establishes new multiplicity results for critical growth anisotropic quasilinear elliptic systems using a novel critical point theorem based on the Z2-cohomological index, applicable to various nonlinear regimes.

Contribution

It introduces a new abstract critical point theorem for symmetric functionals, enabling multiplicity results for critical growth elliptic systems with a unified approach.

Findings

Multiple solutions are proven for different nonlinear regimes based on the parameter gamma.

The new theorem applies to systems with critical growth and is adaptable to nonlocal systems.

The approach broadens the applicability of variational methods in elliptic PDEs.

Abstract

In this paper we prove new multiplicity results for a critical growth anisotropic quasilinear elliptic system that is coupled through a subcritical perturbation term. We identify a certain scaling for the system and a parameter {\gamma} related to this scaling that determines the geometry of the associated variational functional. This leads to a natural classification of different nonlinear regimes for the system in terms of scaling properties of the perturbation term. We give three different types of multiplicity results in the three regimes {\gamma} = 1, {\gamma} > 1, and {\gamma} < 1. Proofs of our multiplicity results are based on a new abstract critical point theorem for symmetric functionals on product spaces, which we prove using the piercing property of the Z2-cohomological index of Fadell and Rabinowitz. This abstract result only requires a local (PS) condition and is therefore applicable to systems with critical growth. It is of independent interest as it has wide applicability to many different types of critical elliptic systems. We also indicate how it can be applied to obtain similar multiplicity results for nonlocal systems.