Vanishing property of principal eigenfunction for cooperative elliptic systems

It is well-known that(see e.g. Proposition 1.3.17 in [10]) the normalized principal eigenfunction $ \phi_1 $ of single elliptic eigenvalue problem$ -d\Delta \phi -c(x)\phi = \lambda\phi $with Robin boundary condition converges to $ 0 $ as $ d\rightarrow 0^+ $ locally uniformly in $ \{x\in\bar\Omega|\; c(x)<\max_{\bar{\Omega}} c(x)\} $. The method used in [10] is designed for single equation and seems difficult to be applied to the elliptic systems directly. In this paper, we extend the conclusion above to elliptic system by introducing a different approach.