Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

We consider fully coupled cooperative systems on \begin{document}$ \mathbb{R}^n $\end{document} with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on \begin{document}$ \mathbb{R}^2 $\end{document} with exponential nonlinearity, are nondegenerate.