Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity

We prove compactness properties along the branches of semi-stable and unstable solutions –whose Morse index is at most one– of the Dirichlet boundary value problem −∆u = λf(x) (1−u)2 on a bounded domain Ω ⊂ R , provided 1 ≤ N ≤ 7. As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. We also show that the result however does not hold for dimension 8 and beyond.