Nonlinear elliptic systems involving Hardy-Sobolev Criticalities

TL;DR

This paper investigates the existence of solutions for nonlinear elliptic systems with Hardy potentials and critical Sobolev exponents, generalizing models like Gross-Pitaevskii and Bose-Einstein systems using variational methods.

Contribution

It introduces a comprehensive variational framework to establish existence results for a broad class of coupled elliptic systems with Hardy-Sobolev criticalities.

Findings

Existence of ground and bound states depending on coupling parameters.

Solutions characterized as minimizers or Mountain-Pass critical points.

Applicable to a wide range of parameters and exponents.

Abstract

This paper is focused on the solvability of a family of nonlinear elliptic systems defined in $\mathbb{R}^N$. Such equations contain Hardy potentials and Hardy-Sobolev criticalities coupled by a possible critical Hardy-Sobolev term. That problem arises as a generalization of Gross-Pitaevskii and Bose-Einstein type systems. By means of variational techniques, we shall find ground and bound states in terms of the coupling parameter $\nu$ and the order of the different parameters and exponents. In particular, for a wide range of parameters we find solutions as minimizers or Mountain-Pass critical points of the energy functional on the underlying Nehari manifold.