Abstract: Firstly, the author discussed the existence of radial positive solutions for a semipositone elliptic equation with the Dirichlet boundary condition by using the variational method, the results show that when the λ is sufficiently small, the equation has no nonnegative solution; when the λ is sufficiently large, the equation has a radial positive solution. Secondly, the author prove that the linearized operator at each solution of the equation has the nonnegative first eigenvalue. Where Ω is a ball or an annulus, the parameter λ>0, f∈C(［0,∞),R) and f(0)<0 (semipositone), k: ［a,b］→［0,∞) and k(|x|) is not always 0. Furthermore, when the Ω is a ball, the k is a linear map; when Ω is an annulus, the k is a monotone increasing function.

Key words: elliptic equation, semipositone problem, radial solution, variational method