Asymptotic formula for eigenvalues of simple pendulum problems

Article type: Research Article

Authors: Shibata, Tetsutaro

Affiliations: The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi‐Hiroshima, 739‐8521, Japan E‐mail: [email protected]‐u.ac.jp

Abstract: We consider the nonlinear eigenvalue problem −Δu=λf(u), u>0 in BR, u=0 on ∂BR, where BR is a ball with radius R>0 and λ>0 is a parameter. Under the appropriate conditions of f, it is known that for a given 0<ε<1, there exists (λ,u)=(λ(ε),uε) satisfying the equation with ∫BRF(uε(x)) dx=|BR|F(u0)(1−ε), where F(u)=∫0uf(s) ds and u0>0 is the smallest zero of f in R+. Furthermore, uε(x)→u0 (x∈BR) and λ(ε)→∞ as ε→0. This concept of parametrization of solution pair by a new parameter ε is based on the variational structure of the equation. We establish the asymptotic formulas for λ(ε) as ε→0 with the ‘optimal’ estimate of the second term.

Keywords: asymptotic formula, variational characterization of solution, simple pendulum

Journal: Asymptotic Analysis, vol. 40, no. 1, pp. 83-91, 2004