Morse theory for a fourth order elliptic equation with exponential nonlinearity

Abstract

Given a Hilbert space \({(\mathcal H,\langle\cdot,\cdot\rangle)}\), and interval \({\Lambda\subset(0,+\infty)}\) and a map \({K\in C^2(\mathcal H,\mathbb R)}\) whose gradient is a compact mapping, we consider the family of functionals of the type: $$I(\lambda,u)=\dfrac12\langle u,u\rangle-\lambda K(u),\quad (\lambda,u)\in\Lambda\times\mathcal H.$$ 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 Poincare–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.