A new critical curve for a class of quasilinear elliptic systems

TL;DR

This paper establishes a new critical curve for quasilinear elliptic systems involving operators like the p-Laplacian and mean curvature, determining conditions for trivial solutions in Euclidean and Carnot group contexts.

Contribution

It introduces a novel critical curve that delineates the existence of nontrivial solutions for a broad class of quasilinear systems, extending previous results to more general operators and settings.

Findings

Systems with exponents below the critical curve have only trivial solutions.

Results apply to both Euclidean space and Carnot groups.

Regularity assumptions ensure the validity of the results.

Abstract

We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.