ln Li and Zhou (SIAM J Sci Comput 23:840-865, 2001), a minimax method for computing multiple solutions of semilinear elliptic equation by calculating critical points of its energy function is presented. But, the method is slow and can find limited amount of solutions. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the energy function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more solutions of semilinear elliptic equation. Numerical experiment confirms these two progresses. In addition, the MP-Newton method inherits the advantages of the minimax method. Finally, a convergence result for the method is established.
