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

We consider the functional $I_{\delta ,{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2}:X\to \mathbb{R}$ 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 $\mathrm{\Omega }$ is a smooth bounded domain of ${\mathbb{R}}^N$, ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2:{\mathbb{R}}^N\to \mathbb{R}$ are convex functions satisfying suitable conditions for $1<p,q<N$, and the nonlinearity $H$ is allowed to exhibit both subcritical and critical growth and may depend on a parameter $\delta \in I\subseteq \mathbb{R}$. Here, the functional space setting is given by the product space $X:=W_0^{1,p}(\mathrm{\Omega })\times W_0^{1,q}(\mathrm{\Omega })$ equipped with the norm $\mathrm{\parallel }z\mathrm{\parallel }=\mathrm{\parallel }u{\mathrm{\parallel }}_{1,p}+\mathrm{\parallel }v{\mathrm{\parallel }}_{1,q}$ for any $z=(u,v)\in X$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{1,s}$ is the usual norm in $W_0^{1,s}(\mathrm{\Omega })$. In this paper we prove that $I_{\delta ,{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2}^{\mathrm{\prime }}$ is of class $(S{)}_{+}$ and we infer that each isolated critical point of $I_{\delta ,{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2}$ has critical groups of finite type and a Poincaré-Hopf formula holds.