Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula

We consider the functional $I_{δ,Ψ_1,Ψ_2}:X \to \mathbb{R}$ defined for any $z=(u,v) \in X$ as \begin{align*} I_{δ,Ψ_1,Ψ_2}(z) & = \int_Ω Ψ_1(\nabla u) \, dx + \int_Ω Ψ_2(\nabla v ) \, dx & - \int_Ω H(δ,x,u(x),v(x)) \,dx, \end{align*} where $Ω$ is a smooth bounded domain of $\mathbb{R}^N$, $