Abstract and Applied Analysis
Volume 2003 (2003), Issue 11, Pages 671-684
doi:10.1155/S1085337503212045
Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems
The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan
Received 17 January 2002
Copyright © 2003 Tetsutaro Shibata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We study the nonlinear two-parameter problem −u″(x)+λu(x)q=μu(x)p, u(x)>0, x∈(0,1), u(0)=u(1)=0. Here, 1<q<p are constants and λ,μ>0 are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p=(3q−1)/2 and this kind of criticality is new for two-parameter problems.