Asymptotics of variational eigenvalues of non-linear Sturm-Liouville problems with two parameters on general level sets

We consider the non-linear Sturm-Liouville problem with two parameters on general level set N_α: $\begin{equation}\left\{\begin{array}{l@{\quad}l}-u''(x)=\mu u(x)-\lambda(u(x)+|u(x)|^{p-1}u(x)),&x\in I=(0,1),\\u(0)=u(1)=0,&\end{array}\right.\end{equation}$ where p>1, μ,λ∈R are parameters and $\begin{equation}N_{\alpha}:=\Biggl\{u\in W_{0}^{1,2}(I){:}\ \int_{0}^{1}(u'(x)^{2}-\mu u(x)^{2})\,\mathrm{d}x=2\alpha,\ \alpha <0{:}\ \mbox{is a normalizing parameter}\Biggr\}.\end{equation}$ We establish an asymptotic formula of n-th variational eigenvalue λ=λ_n(α) of (1) as α→−∞:

λ_n(α)=C_μ(−α)^(1−p)/2+o((−α)^(1−p)/2).

Furthermore, we give an asymptotic formula of C_μ as μ→∞.