In this current work, we examine a \((p_1(x),\ldots ,p_n(x))\) -Laplacian system with Hardy potentials. We establish the existence of at least one non-zero critical point and at least three distinct critical points to this system from an abstract critical point result of Bonanno et al. (Adv Nonlinear Stud 14(4):915–939, 2014) and a recent three critical points theorem of Bonanno and Marano (Appl Anal 89:1–10, 2010). This article represents, as far as we are aware, one of the first efforts towards the study of the \((p_1(x),\ldots ,p_n(x))\) -Laplacian systems with Hardy potentials, in which the nonlinearity in \(\Omega \) may change sign. Furthermore, we provide an example to illustrate our main conclusions.
