A Poincaré–Hopf formula for functionals associated to quasilinear elliptic systems

• Published in: Nonlinear analysis: real world applications Vol. 87; p. 104443
• Main Authors: Borgia, Natalino, Cingolani, Silvia, Vannella, Giuseppina
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2026
• Subjects: Banach spaces; Critical groups; Critical growth; Degree theory; Poincaré–Hopf formula; Quasilinear elliptic systems; 58E05; Quasilinear elliptic systems; 35J62; 47H11; 35J50; Critical groups; Poincaré–Hopf formula; Critical growth; Banach spaces; Degree theory
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2025.104443

We consider the functional Jα,β(z)=1p∫Ωα+|∇u(x)|2p2dx+1q∫Ωβ+|∇v(x)|2q2dx−∫ΩF(u(x),v(x))dx,z=(u,v)∈X, where Ω is a smooth bounded domain of RN, 1<p,q<N, α,β≥0. Here X≔W01,p(Ω)×W01,q(Ω) denotes the product space, endowed with the norm ‖z‖=‖u‖1,p+‖v‖1,q, for any z=(u,v)∈X, being ‖⋅‖1,s the usual norm in W01,s(Ω). In this paper we prove that Jα,β′ is of class (S)+ and, from Cingolani and Degiovanni (2009), Theorem 1.1, we infer that each isolated critical point of Jα,β has critical groups of finite type and a Poincaré–Hopf formula holds.