Representation and asymptotic formulas for 1-dimensional linearized eigenvalue problems with Dirichlet boundary condition Original Research Article

In this article nontrivial solutions to a nonlinear Dirichlet boundary value problem with a small parameter and the corresponding linearized eigenvalue problems are investigated. In a typical case of nonlinearity, asymptotic formulas of all eigenvalues and asymptotic profiles of all eigenfunctions are proved. Proofs are based on the method of representation equation and asymptotic results in the corresponding Neumann problem obtained in the recent papers by the author and S. Yotsutani.