A Poincaré–Hopf formula for functionals associated to quasilinear elliptic systems - Borgia, Natalino; Cingolani, Silvia; Vannella, Giuseppina (Nonlinear analysis: real world applications, 2026)
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.