On a system involving a critically growing nonlinearity

This paper deals with the system \{{array}{ll} -\Delta u = \lambda u + q |u|^3 u \phi & \hbox{in} B_R, -\Delta \phi=q |u|^5 & \hbox{in} B_R, u=\phi=0 & \hbox{on} \partial B_R. {array}. We prove existence and nonexistence results depending on the value of $\lambda$.