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

Abstract In this paper we study a class of Hardy–Sobolev type systems defined in $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> 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.