This paper presents necessary and sufficient conditions for the existence of a Lyapunov function in the Lur'e form that guarantees the absolute stability of a Lur'e control system with multiple nonlinearities. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs). If these inequalities are feasible, the free parameters in the Lyapunov function, such as the positive definite matrix and the coefficients of the integral terms,are given by the solution of the LMIs. Otherwise, there does not exist any Lyapunov function in the Lur'e form. Necessary and sufficient conditions are also obtained for the robust absolute stability of time-varying structured uncertain systems.