Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems. (English) Zbl 0894.34016

Shibata, T.

Differ. Integral Equ. 10, No. 4, 625-648 (1997).

The vector problem under consideration reads: \[ u''+\sum _{k=1}^n u_kf_k(u)=\lambda g(u),\;u(x)>0\;\text{ for } x\in (0,1),\;u(0)=u(1)=0, \] where \(u_k\), \(k=1,\dots ,n,\) and \(\lambda \) are positive parameters.
Spectral asymptotics are established for the variational eigenvalues in the frame of the Ljusternik-Schnirelman theory on general level sets elaborated by E. Zeidler [Math. Nachr. 129, 235-259 (1986; Zbl 0608.58014)].

Reviewer: J.Andres (Olomouc)

MSC:

34B24	Sturm-Liouville theory
34B15	Nonlinear boundary value problems for ordinary differential equations
34L20	Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators

Keywords:

multiparameter Sturm-Liouville problems; asymptotic formulas

Citations:

Zbl 0608.58014