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

We consider the functional 𝐽_(𝛼,𝛽) (𝑧) = 1/𝑝 ∫_𝛺 (𝛼 + |∇𝑢(𝑥)|^2) ^(𝑝/2) 𝑑𝑥 + 1/𝑞 ∫_𝛺(𝛽 + |∇𝑣(𝑥)|^2)^(𝑞/2) 𝑑𝑥 − ∫_𝛺 𝐹 (𝑢(𝑥), 𝑣(𝑥)) 𝑑𝑥, 𝑧 = (𝑢, 𝑣) ∈ 𝑋, where 𝛺 is a smooth bounded domain of R^𝑁 , 1 < 𝑝, 𝑞 < 𝑁, 𝛼, 𝛽 ≥ 0. Here 𝑋 ∶= (𝑊_0)^(1,𝑝) (𝛺)×(𝑊_0)^(1,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 (𝑊_0)^(1,𝑠) (𝛺). In this paper we prove that 𝐽_(𝛼,𝛽) ′ is of class (𝑆)+ and, from Cingolani and Degiovanni (2009), Theorem 1.1, we infer that each isolated critical point of 𝐽_(𝛼,𝛽) has critical groups of finite type and a Poincaré-Hopf formula holds.