Characterizing the existence of coexistence states in a class of cooperative systems (Q2380881)

The authors discussed the existence of coexistence state in a cooperative system which is an elliptic boundary value problem (BVP) for \((u,v)\) with Dirichlet condition: \(-\Delta u=\lambda u+\alpha u-a(x)f(x,u)\), \( -\Delta u=\beta u+\lambda v\) for \(x\in \Omega\). Let \(\sigma_1>0\) be the principal eigenvalue of \(-\Delta\), they obtain that this BVP possesses a coexistence state if and only if \((\sigma_1-\lambda)^2<\alpha \beta<\Sigma(\lambda)\), where \(\Sigma(\lambda)\) is some number depending on \(\Delta, \lambda\) and a sub-domain \(\Omega_0\subset\Omega\).