Existence of solutions for a system with general Hardy--Sobolev singular criticalities

TL;DR

This paper proves the existence of positive solutions for a class of Hardy--Sobolev systems with independent and wide-range singularities using variational methods and critical point theory.

Contribution

It introduces a novel analysis of Hardy--Sobolev systems with independent singularity orders and establishes existence results via variational techniques.

Findings

Existence of positive bound states.

Existence of ground states.

Solutions found as minimizers or Mountain-Pass points.

Abstract

In this paper we study a class of Hardy--Sobolev type systems defined in $\mathbb{R}^N$ and coupled by a singular critical Hardy--Sobolev term. The main novelty of this work is that the orders of the singularities are independent and contained in a wide range. By means of variational techniques, we will prove the existence of positive bound and ground states for such a system. In particular, we find solutions as minimizers or Mountain--Pass critical points of the energy functional on the underlying Nehari manifold.