On Finding Multiple Solutions to a Singularly Perturbed Neumann Problem

In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the numerical results, a Morse index approach is used to identify the Morse index of the root solution uε = 1 at any perturbation value, its bifurcation points and then the critical perturbation value. Many interesting numerical solutions are computed for the first time and displayed with their contours and mesh profiles to illustrate the theory and method.