Compactness of a nonlinear eigenvalue problem with a singular nonlinearity

We study the Dirichlet boundary value problem −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ RN. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ ∗ in the class of all solutions with finite Morse index, λ ∗ being the extremal value associated to the nonlinear eigenvalue problem