Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space (H, 〈.,.〉) and interval λ ⊂(0,+∞) and a map K ∈ C²(H,ℝ) whose gradient is a compact mapping, we consider the family of functionals of the type: I(λ,u)=12;〈u,u〉 - λK(u), (λ,u) ∈ λ × H. As already observed by many authors, for the functionals we are dealing with the (PS) condition may fail under just this assumptions. Nevertheless, by using a recent deformation Lemma proven by Lucia (Topol Methods Nonlinear Anal 30(1):113-138, 2007), we prove a Poincaré-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for a fourth order nonlinear scalar field equation on a bounded and smooth C^∞ region of the four dimensional Euclidean space in the flavor of (Malchiodi in Adv Differ Equ 13:1109-1129, 2008). We remark that this formula has been proven with complete different methods in (Lin and Wei Preprint, 2007) by using blow-up type estimates.