Spectral asymptotics and bifurcation for nonlinear multiparameter elliptic eigenvalue problems. (English) Zbl 0861.35071

Shibata, Tetsutaro

Bull. Belg. Math. Soc. - Simon Stevin 3, No. 5, 501-515 (1996).

Summary: This paper is concerned with the nonlinear multiparameter elliptic eigenvalue problem \[ u''(r)+{N-1\over r} u'(r)+\mu u(r)-\sum^k_{i=1} \lambda_if_i(u(r))=0,\quad 0<r<1, \] \[ u(r)>0,\quad 0\leq r<1,\quad u'(0)=0,\quad u(1)=0, \] where \(N\geq 1\), \(k\in\mathbb{N}\) and \(\mu,\lambda_i\geq 0\) \((1\leq i\leq k)\) are parameters. The aim of this paper is to study the asymptotic properties of eigencurve \(\mu(\lambda,\alpha)= \mu(\lambda_1,\lambda_2,\dots,\lambda_k,\alpha)\) with emphasis on the phenomenon of bifurcation from the first eigenvalue \(\mu_1\) of \(-\Delta|_D\) and on gaining a clearer picture of the bifurcation diagram. Here, \(\alpha>0\) is a normalizing parameter of the eigenfunction associated with \(\mu(\lambda,\alpha)\). To this end, we establish asymptotic formulas of \(\mu(\lambda,\alpha)\) as \(|\lambda|\to\infty, 0\).

MSC:

35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
34B15	Nonlinear boundary value problems for ordinary differential equations
35P20	Asymptotic distributions of eigenvalues in context of PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

radial solutions; multiparameter eigenvalue problem; bifurcation from the first eigenvalue