Uniqueness of positive solutions for a class of semipositone elliptic systems (Q858661)

Consider the boundary value problem \[ \begin{gathered} -\Delta u=\lambda f(v),\quad -\Delta v= \mu g(u)\quad\text{for }x\in \Omega,\\ u= 0= v\quad\text{for }x\in \partial\Omega,\end{gathered}\tag{\(*\)} \] where \(\Omega\) is the open unit ball in \(\mathbb{R}^N\), \(N\geq 1\) with smooth boundary \(\partial\Omega\), \(\lambda\) and \(\mu\) are positive parameters. The authors derive conditions on \(f\) and \(g\) such \((*)\) has for sufficiently large \(\lambda\) and \(\mu\) a unique positive solution. The proof is based on a boundary value problem for an ODE-system.