Asymptotic formulas of the eigenvalues for the linearization of the scalar field equation

Abstract

We establish asymptotic formulas for all the eigenvalues of the linearization problem of the Neumann problem for the scalar field equation in a finite interval \[ \begin{cases} \varepsilon^2u_{xx}-u+u^3=0, & 0< x<1,\\ u_x(0)=u_x(1)=0. \end{cases} \] In the previous paper of the third author [T. Wakasa and S. Yotsutani, J. Differ. Equ. 258 (2015), 3960–4006] asymptotic formulas for the Allen–Cahn case $\varepsilon ^2u_{xx}+u-u^3=0$ were established. In this paper, we apply the method developed in the previous paper to our case. We show that all the eigenvalues can be classified into three categories, i.e., near $-3$ eigenvalues, near $0$ eigenvalues and the other eigenvalues. We see that the number of the near $-3$ eigenvalues (resp. the near $0$ eigenvalues) is equal to the number of the interior and boundary peaks (resp. the interior peaks) of a solution for the nonlinear problem. The main technical tools are various asymptotic formulas for complete elliptic integrals.