Precise spectral asymptotics for nonlinear Sturm-Liouville problems
Given a constant \(p>1\), it is known from the work of \textit{H. Berestycki} [J. Funct. Anal. 40, 1-29 (1981; Zbl 0452.35038)] and others that, for each \(\alpha >0\), there is a unique solution \((\lambda ,u)=(\lambda (\alpha),u_{\alpha})\in {\mathbb{R}}_+ \times C^2 (\bar{I})\) of \(-u'' +u^p =\lambda u\) on \(I:=(0,1)\) satisfying \(u_{\alpha}>0\) on \(I\), \(u_{\alpha}(0)=u_{\alpha}(1)=0\) and \(\|u_{\alpha}\|_2 =\alpha\). This paper establishes the asymptotic expansions of \(\lambda (\alpha)\), \(u' _{\alpha}(0)^2\) and \(\|u_{\alpha}\|_{\infty}^{p-1}\) as \(\alpha \to \infty\). As a corollary, some analogous results are proved for \(-u''+|u|^{p-1}u=\lambda u\) on \(I\), \(u(0)=u(1)=0\), without the sign restriction.
scientific article; zbMATH DE number 1750349