July  2008, 7(4): 745-763. doi: 10.3934/cpaa.2008.7.745

Representation formulas for some 1-dimensional linearized eigenvalue problems

1. 

Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194

Received  February 2007 Revised  December 2007 Published  April 2008

We study nontrivial stationary solutions to a nonlinear boundary value problem with parameter $\varepsilon>0$ and the corresponding linearized eigenvalue problem. By using a particular solution of a linear ordinary differential equation of the third order, we give expressions of all eigenvalues and eigenfunctions to the linearized problems. They are completely determined by a characteristic function which consists of complete elliptic integrals. We also show asymptotic formulas of eigenvalues with respect to sufficiently small $\varepsilon$. These results give important information for profiles of corresponding eigenfunctions.
Citation: Tohru Wakasa, Shoji Yotsutani. Representation formulas for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 745-763. doi: 10.3934/cpaa.2008.7.745
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