International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 679-688
doi:10.1155/S0161171296000944

Deformation of domain and the limit of the variational eigenvalues of semilinear elliptic operators

Tetsutaro Shibata

Department of Mathematics, Aichi Prefectural University 3-28, Takada-cho, Mizuho-ku, Nagoya 467, Japan

Received 11 April 1994; Revised 30 August 1994

Copyright © 1996 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 consider the semilinear elliptic eigenvalue problem {Lu+f(x,u)=μu  in  Ωr(r≥0),u=0  on  ∂Ωr. The asymptotic behavior of the variational eigenvalues μ=μn(r,α) obtained by Ljusternik-Schnirelman theory is studied when the domain Ω0 is deformed continuously. We also consider the cases that Vol(Ωr)→0,∞ as r→∞.