题目：Solving Singularly Perturbed Neumann Problems for Multiple Solutions
In this talk, based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value, the generating process of nontrivial positive solutions for a general singularly perturbed Neumann boundary value problem is developed. The bifurcation points of each trivial solution and then the exact critical perturbation value $\varepsilon_c$ which determines the existence or non-existence of nontrivial positive solutions are verified. An efficient local minimax method based on the bifurcation and Morse theory is proposed to compute both M-type and W-type saddle points by introducing an adaptive local refinement strategy, a continuation strategy for initial selection and the Newton method to improve the convergence speed. Extensive numerical results are reported to investigate the critical value $\varepsilon_c$ and present interesting properties of different types of multiple solutions.