Weak solutions of quasilinear elliptic eystems via the cohomological index

Abstract

In this paper we study a class of quasilinear elliptic systems of the type $$ \begin{cases} -{\rm div}(a_1(x,\nabla u_1,\nabla u_2))=f_1(x,u_1,u_2) & \text{in } \Omega,\\ -{\rm div}(a_2(x,\nabla u_1,\nabla u_2))=f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \end{cases} $$ with $\Omega$ bounded domain in $\mathbb R^N$. We assume that $A\colon \Omega \times {\mathbb{R}}^N\times{\mathbb{R}}^N\rightarrow{\mathbb{R}}$, $F\colon \Omega \times {\mathbb{R}} \times {\mathbb{R}} \rightarrow {\mathbb{R}}$ exist such that $a=(a_1,a_2)=\nabla A$ satisfies the so called Leray-Lions conditions and $f_1={\partial F}/{\partial u_1}$, $f_2={\partial F}/{\partial u_2}$ are Carathéodory functions with subcritical growth.

The approach relies on variational methods and, in particular, on a cohomological local splitting which allows one to prove the existence of a nontrivial solution.