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

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→∞.