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

Abstract

<p style='text-indent:20px;'>Concerning a class of diffusive logistic equations, Ni [<xref ref-type="bibr" rid="b1">1</xref>,Abstract] proposed an optimization problem to consider the supremum of the ratio of the <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> in the one-dimensional case. In [<xref ref-type="bibr" rid="b1">1</xref>], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an <inline-formula><tex-math id="M3">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> unbounded sequence of sub-solutions.