A global compactness result for a mixed boundary value problem in half cylinder

Let ΩΩ be a bounded smooth domain in RNRN with N≥2N≥2, C=Ω×(0,∞)C=Ω×(0,∞) and {un}⊂H0,L1(C) be a (P.S.) sequence of the functional Iλ(u)=12∫C∣∇u∣2dxdy−λ2∫Ω×{0}∣u∣2dx−N−12N∫Ω×{0}∣u∣2NN−1dx,λ∈R. We study the limit behaviour of unun and obtain a global compactness result.