Abstract

We consider the semilinear elliptic eigenvalue problem {Lu+f(x,u)=μuinΩr(r0),u=0onΩr. 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.