S-shaped bifurcation curves for nonlinear two-parameter problems

We consider the nonlinear eigenvalue problem −u″(t)=λ(1+u(t)+u(t)2−ϵu(t)p),u(t)>0,t∈I:=(−1/2,1/2),u(−1/2)=u(1/2)=0,u(−1/2)=u(1/2)=0, where p>2p>2 is a constant, λ>0λ>0 and ϵ>0ϵ>0 are parameters. This equation was introduced by Crandall and Rabinowitz (1973) as the typical model which develops the S-shaped bifurcation curve when p=3p=3 and 0<ϵ≪10<ϵ≪1. It is known that if p>2p>2 and 0<ϵ≪10<ϵ≪1, then there exists a unique solution (λϵ(α),uϵ,α)∈R+×C2(Ī) with ‖uϵ,α‖∞=α‖uϵ,α‖∞=α, where α>0α>0 is a given constant. Furthermore, λ=λϵ(α)λ=λϵ(α) is S-shaped. In this paper, we establish the asymptotic formulas for λ=λϵ(α)λ=λϵ(α) as α→0,α1,ϵ,α2,ϵ,βϵα→0,α1,ϵ,α2,ϵ,βϵ when 0<ϵ≪10<ϵ≪1, where α1,ϵα1,ϵ and α2,ϵα2,ϵ are two turning points of λϵ(α)λϵ(α), and βϵβϵ is the unique positive solution to the algebraic equation 1+x+x2−ϵxp=01+x+x2−ϵxp=0. We also characterize the asymptotic shape of uϵ,α(t)uϵ,α(t) as α→βϵα→βϵ by establishing the asymptotic behavior of ‖uϵ,α‖q‖uϵ,α‖q as α→βϵα→βϵ when 0<ϵ≪10<ϵ≪1 is a fixed constant.