Optimal control problems governed by degenerate quasi-linear partial differential equations of elliptic type are considered. The optimal control systems considered may lack Cesari-type conditions, and therefore the corresponding approximate optimal control problem may have no solution. To yield the maximum principle of optimal pairs, relaxed controls are used to overcome the difficulties occurring when considering approximate problems. The relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated.