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Concerning a class of diffusive logistic equations, Ni [1, Abstract] proposedan optimization problem to consider the supremum of the ratio of the L^1 normsof species and resources by varying the diffusion rates and the profiles ofresources, and moreover, he gave a conjecture that the supremum is 3 in theone-dimensional case. In [1], Bai, He and Li proved the validity of thisconjecture. The present paper shows that the supremum is infinity in a casewhen the habitat is a multi-dimensional ball. Our proof is based on thesub-super solution method. A key idea of the proof is to construct an L^1unbounded sequence of sub-solutions.

On the unboundedness of the ratio of species and resources for the diffusive logistic equation