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We consider the stationary problem for a diffusive logistic equation with thehomogeneous Dirichlet boundary condition. Concerning the corresponding Neumannproblem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio ofthe total masses of species to resources. For this question, Bai, He and Lishowed that the supremum of the ratio is 3 in the one dimensional case, and theauthor and Kuto showed that the supremum is infinity in the multi-dimensionalball. In this paper, we show the same results still hold true for the Dirichletproblem. Our proof is based on the sub-super solution method and needs moredelicate calculation because of the range of the diffusion rate for theexistence of the solution.

On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition