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
Abatangelo, L., & Portaluri, A. (2011). Morse theory for a fourth order elliptic equation with exponential nonlinearity. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 18(1), 27-43 [10.1007/s00030-010-0082-1].
Citazione: | Abatangelo, L., & Portaluri, A. (2011). Morse theory for a fourth order elliptic equation with exponential nonlinearity. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 18(1), 27-43 [10.1007/s00030-010-0082-1]. | |
Tipo: | Articolo in rivista - Articolo scientifico | |
Carattere della pubblicazione: | Scientifica | |
Presenza di un coautore afferente ad Istituzioni straniere: | No | |
Titolo: | Morse theory for a fourth order elliptic equation with exponential nonlinearity | |
Autori: | Abatangelo, L; Portaluri, A | |
Autori: | ||
Data di pubblicazione: | 2011 | |
Lingua: | English | |
Rivista: | NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS | |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s00030-010-0082-1 | |
Appare nelle tipologie: | 01 - Articolo su rivista |