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

### Weak solutions of quasilinear elliptic systems via a cohomological index

#### Abstract

In this paper we study a class of quasilinear elliptic systems of the type $\left\{\begin{array}{ll} - \divg(a_1(x,\nabla u_1,\nabla u_2))\ =\ f_1(x,u_1,u_2) & \text{in } \Omega,\\ - \divg(a_2(x,\nabla u_1,\nabla u_2))\ =\ f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \end{array}\right.$ with $\Omega$ bounded domain in $\R^N$. We assume that $A : \Omega \times \mathbb{R}^N \times \mathbb{R}^N \rightarrow \mathbb{R}$, $F : \Omega \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ exist such that $a=(a_1,a_2)=\nabla A$ satisfies the so called Leray-Lions conditions and $f_1=\frac{\partial F}{\partial u_1}$, $f_2=\frac{\partial F}{\partial u_2}$ are Carathéodory functions with {\sl 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/13568
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