Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
Tohru Wakasa, Shoji Yotsutani · Communications on Pure & Applied Analysis · 2010
We are interested in the asymptotic profiles ofall eigenfunctions for 1-dimensionallinearized eigenvalue problemsto nonlinear boundary value problemswith a diffusion coefficient $\varepsilon$.For instance, it seems thatthey have simple and beautiful propertiesfor sufficiently small $\varepsilon$in the balanced bistable nonlinearity case.As the first step to give rigorous proofsfor the above case,we study the case $f(u)=\sin u$ precisely.We show that two special eigenfunctions completelycontrol the asymptotic profiles of other eigenfunctions.