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

Abstract

We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li [1] showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto [14] showed that the supremum is infinity in the multi-dimensional ball. In this paper, we prove that the same results still hold true for the Dirichlet problem, and our proof is based on the sub-super solution method. Moreover, we show the differences of the solutions corresponding to the maximizing sequence between the one- and two-dimensional domains.