A Poincar\'e-Hopf formula for functionals associated to quasilinear elliptic systems

Abstract

We consider the functional \begin{align*} J_{\alpha,\beta}(z) =& \frac{1}{p} \into \left(\alpha+ |\nabla u(x)|^2\right)^{\frac p 2} \ dx + \frac{1}{q}\into \left(\beta + |\nabla v(x)|^2\right)^{\frac q 2} \ dx \nonumber \\ &- \into F(u(x),v(x)) \ dx, \quad z=(u,v) \in X, \end{align*} where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $