Concentration phenomena for Liouville's equation in dimension four

Let Ω be a bounded domain of R and let uk be solutions to the equation (1) ∆uk = Vke in Ω, where (2) Vk → 1 uniformly in Ω, as k → ∞. Throughout the paper we denote as ∆ = − ∑ i( ∂ ∂xi ) 2 the Laplacian with the geometers’ sign convention. Continuing the analysis of [18], here we study the compactness properties of equation (1). Equation (1) is the fourth order analogue of Liouville’s equation. Thus, for problem (1), (2) we may expect similar results to hold as have been obtained by Brezis-Merle [3] in the two-dimensional case. Recall the following result from [3] (we also refer to Li-Shafrir [11]). Theorem 1.1. Let Σ be a bounded domain of R and let (uk)k∈N be a sequence of solutions to the equation (3) ∆uk = Vke in Σ, where Vk → 1 uniformly in Σ as k →∞, and satisfying the uniform bound