On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Abstract

Based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value for a general semilinear singularly perturbed Neumann boundary value problem, in this paper, the exact critical perturbation value $\varepsilon_c$ which determines the existence or nonexistence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.