On a system involving a critically growing nonlinearity

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