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→∞.