We consider the non-linear Sturm-Liouville problem with two parameters on general level set Nα:
$\begin{equation}\left\{\begin{array}{l@{\quad}l}-u''(x)=\mu u(x)-\lambda(u(x)+|u(x)|^{p-1}u(x)),&x\in I=(0,1),\\u(0)=u(1)=0,&\end{array}\right.\end{equation}$
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=λn(α) of (1) as α→−∞:
λn(α)=Cμ(−α)(1−p)/2+o((−α)(1−p)/2).
Furthermore, we give an asymptotic formula of Cμ as μ→∞.
