Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity

We study a semilinear Dirichlet problem where the nonlinearlity is monotone, negative at zero (Semipositone), and concave. We assume that the outer boundary of the region is convex and show that for large values of a parameter there can be only one non-negative solution. We show that such a solution is positive in the region.