一类奇异摄动Neumann问题变号解的计算与性质

Computation and Properties of Sign-Changing Solutions for a Singularly Perturbed Neumann Problem

采用改进型局部极小极大方法(LMM)计算一类奇异摄动半线性椭圆Neumann问题中的多种变号解。一方面,按照Morse指标由低到高的顺序计算模型问题在不同非线性与不同区域情形下的变号解;另一方面,模拟模型问题的同一类变号解的能量与最大值随奇异摄动参数的变化规律。严格证明了变号解的最大值关于奇异摄动参数的非一致有界性,及变号解的Morse指标在一定条件下大于等于2的结论。基于大量的数值试验,提出一个关于同一类变号解能量和最大值的猜想。

An improved local minimax method(LMM)has been adopted for the calculation of sign-changing solutions for a class of singularly perturbed Semilinear Elliptic Neumann problems.On the one hand,the variable signchanging solutions of the model problem with different nonlinear terms and domains can be achieved in the ascending order of Morse indices.On the other hand,the energy and maximum value of the same class of variable number solution of the model problem are changing in line with the variation of the singular perturbation parameter.It can be strictly proved that the maximum value of the sign-changing solution is characterized with nonuniform boundedness with respect to the singularly perturbed parameter,with the Morse index of the sign-changing solution equal to or greater than 2 under certain conditions.Based on extensive numerical experiments,a conjecture has thus been proposed about the energy and maximum for the same kind of sign-changing solutions.