\(L^q\) spectral asymptotics for nonlinear Sturm-Louiville problems. (English) Zbl 1212.34273

Shibata, Tetsutaro

Differ. Integral Equ. 19, No. 7, 773-783 (2006).

Summary: We consider the nonlinear Sturm-Liouville problem \[ -u''(t)+f(u(t))=\lambda u(t),\;u(t)>0,\;t\in I:=(0,1),\;u(0)=u(1)=0, \] where \(\lambda >0\) is an eigenvalue parameter. For a better understanding of the global behavior of the branch of positive solutions in \(\mathbb R_+\times L^q(I)\) (\(1\leq q\leq \infty \)), we establish precise asymptotic formulas for the eigenvalue \(\lambda \) with respect to \(\| u_\lambda \| _q\), where \(u_{\lambda }\) is the unique solution associated with a given \(\lambda >\pi ^2\).

MSC:

34L20	Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B24	Sturm-Liouville theory

Keywords:

nonlinear Sturm-Liouville problem; positive solution; global behavior