Existence of bound and ground states for an elliptic system with double criticality

TL;DR

This paper investigates the existence of bound and ground states for a class of nonlinear elliptic systems with critical nonlinearities and singular potentials, using variational methods to extend previous results.

Contribution

It introduces a variational approach that broadens the parameter ranges for which existence results are known, including new cases with critical powers and singular potentials.

Findings

Existence of ground states for large or small positive coupling parameters.

Bound states identified as Mountain-Pass critical points on the Nehari manifold.

Improved results covering previously unconsidered parameter ranges.

Abstract

We study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter $\nu$ is large or $\nu$ is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain-Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves some known results and allows us to cover range of parameters which have not been considered previously.