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

Article type: Research Article

Authors: Shibata, Tetsutaro

Affiliations: Department of Mathematics, Aichi Prefectural University, 3–28 Takada-cho, Mizuho-ku, Nagoya 467, Japan

Abstract: 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→∞.

DOI: 10.3233/ASY-1995-10103

Journal: Asymptotic Analysis, vol. 10, no. 1, pp. 49-61, 1995