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

Abstract

We consider the functional $I_{\delta ,{\Psi }_1,{\Psi }_2}:X\to \mathbb{R}$ 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 $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^N$, ${\Psi }_1,{\Psi }_2:{\mathbb{R}}^N\to \mathbb{R}$ 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 $X:=W_0^{1,p}(\Omega )\times W_0^{1,q}(\Omega )$ equipped with the norm $\Vert z\Vert =\Vert u{\Vert }_{1,p}+\Vert v{\Vert }_{1,q}$ for any $z=(u,v)\in X$, where $\Vert \cdot {\Vert }_{1,s}$ is the usual norm in $W_0^{1,s}(\Omega )$. In this paper we prove that $I_{\delta ,{\Psi }_1,{\Psi }_2}^{\prime }$ is of class $(S{)}_{+}$ and we infer that each isolated critical point of $I_{\delta ,{\Psi }_1,{\Psi }_2}$ has critical groups of finite type and a Poincaré-Hopf formula holds.

arXiv commentTo appear on Differential and Integral Equations