Weak solutions of quasilinear elliptic eystems via the cohomological index
Keywords
Quasilinear elliptic system, Leray-Lions conditions, subcritical growth, cohomological index, variational approach, p-Laplacian operatorAbstract
In this paper we study a class of quasilinear elliptic systems of the type $$ \cases -\divg(a_1(x,\nabla u_1,\nabla u_2))=f_1(x,u_1,u_2) & \text{in } \Omega,\\ -\divg(a_2(x,\nabla u_1,\nabla u_2))=f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \endcases $$ with $\Omega$ bounded domain in $\R^N$. 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 {\it 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.Downloads
Published
2010-04-23
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1.
CANDELA, Anna Maria, DE MEDEIROS, Everaldo Souto, PALMIERI, Giuliana and PERERA, Kanishka. Weak solutions of quasilinear elliptic eystems via the cohomological index. Topological Methods in Nonlinear Analysis. Online. 23 April 2010. Vol. 36, no. 1, pp. 1 - 18. [Accessed 28 March 2024].
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