Asymptotic formula for eigenvalues of simple pendulum problems

We consider the nonlinear eigenvalue problem −Δu=λf(u), u>0 in B_R, u=0 on ∂B_R, where B_R 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),u_ε) satisfying the equation with ∫_BRF(u_ε(x)) dx=|B_R|F(u₀)(1−ε), where F(u)=∫₀^uf(s) ds and u₀>0 is the smallest zero of f in R₊. Furthermore, u_ε(x)→u₀ (x∈B_R) 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.