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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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