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

We consider the semilinear elliptic eigenvalue problem { L u + f ( x , u ) = μ u in Ω r ( r ≥ 0 ) , u = 0 on ∂ Ω r . 
The asymptotic behavior of the variational eigenvalues μ = μ n ( r , α ) obtained by Ljusternik-Schnirelman 
theory is studied when the domain Ω 0 is deformed continuously. We also consider the cases that 
 V o l ( Ω r ) → 0 , ∞ as r → ∞ .