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Weak solutions of quasilinear elliptic systems via the cohomological index. (English) Zbl 1217.35061
Summary: We study a class of quasilinear elliptic systems of the type
$\begin{cases} -\operatorname{div}g(a_1(x,\nabla u_1,\nabla u_2))=f_1(x,u_1,u_2) & \text{in } \Omega,\\ -\operatorname{div}g(a_2(x,\nabla u_1,\nabla u_2))=f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \end{cases}$
with $$\Omega$$ bounded domain in $$\mathbb R^N$$. We assume that $$A:\Omega \times \mathbb R^N\times\mathbb R^N\rightarrow\mathbb R$$, $$F:\Omega \times\mathbb R\times\mathbb R\rightarrow \mathbb R$$ exist such that $$a=(a_1,a_2)=\nabla A$$ satisfies the so called Leray-Lions conditions and $$f_1=\partial F/\partial u_1$$, $$f_2=\partial F/\partial u_2$$ 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.

##### MSC:
 35J57 Boundary value problems for second-order elliptic systems 35J50 Variational methods for elliptic systems 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 47J30 Variational methods involving nonlinear operators
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