Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space H, an interval Λ⊂(0,+∞) and a map K∈C2(H,R) whose gradient is a compact mapping, the authors consider a family of functionals of the form I(λ,u)=12⟨u,u⟩−λK(u),(λ,u)∈Λ×H. Based on a recently proven deformation lemma, they show a Poincaré-Hopf-type theorem which they use, together with precise homological properties of the formal set of barycenters, in order to establish a direct and geometrically clear degree counting formula for the following fourth-order nonlinear scalar field equation on bounded, smooth domains of R4: ⎧⎩⎨Δ2u=τh(x)eu∫Ωh(x)eudxu=Δu=0in Ω,on ∂Ω,(1) where h∈C2,α(Ω) is a positive function and τ>0. More specifically, they re-prove the following result proved earlier by other authors using blow-up estimates: Theorem 2. For τ∈(64kπ2,64(k+1)π2) and k∈N, the Leray-Schauder degree dτ of (1) is given by dτ=(k−χ(Ω)k), where χ(Ω) denotes the Euler characteristic of the domain Ω. In particular, if χ(Ω)⩽0 and τ≠64kπ2, then problem (1) has a solution.