We establish conditions for existence and for nonexistence of nontrivial solutions to an elliptic system of partial differential equations. This system is of gradient type and has a nonlinearity with critical growth.

In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, \[\displaylines{(-\Delta_p)^su-\mu \frac{|u|^{p-2}u}{|x|^{sp}}=\lambda u^{-\alpha}+\frac{|u|^{p_s^*(t)-2}u}{|x|^t}, \quad\hbox{in }\Omega, \\ u>0,\quad\text{in }\Omega,\\ \quad u=0, \quad\text{in } \mathbb{R}^N \setminus\Omega }\] where \(\Omega \subset \mathbb{R}^N\) is bounded domain...