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

Abstract

We consider the functional $J_{\alpha ,\beta }\left(z\right)=\frac{1}{p}{\int }_{\Omega }{\left(\alpha +{|\nabla u\left(x\right)|}^2\right)}^{\frac{p}{2}}dx+\frac{1}{q}{\int }_{\Omega }{\left(\beta +{|\nabla v\left(x\right)|}^2\right)}^{\frac{q}{2}}dx-{\int }_{\Omega }F(u\left(x\right),v\left(x\right))dx,z=(u,v)\in X,$ where $\Omega$ is a smooth bounded domain of $R^N$, $1<p,q<N$, $\alpha ,\beta \ge 0$. Here $X≔W_0^{1,p}\left(\Omega \right)\times W_0^{1,q}\left(\Omega \right)$ denotes the product space, endowed with the norm $\Vert z\Vert ={\Vert u\Vert }_{1,p}+{\Vert v\Vert }_{1,q}$, for any $z=(u,v)\in X$, being ${\Vert \cdot \Vert }_{1,s}$ the usual norm in $W_0^{1,s}\left(\Omega \right)$. In this paper we prove that $J_{\alpha ,\beta }^{\prime }$ is of class ${\left(S\right)}_{+}$ and, from Cingolani and Degiovanni (2009), Theorem 1.1, we infer that each isolated critical point of $J_{\alpha ,\beta }$ has critical groups of finite type and a Poincaré–Hopf formula holds.