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May  2021, 26(5): 2441-2450. doi: 10.3934/dcdsb.2020186

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

 1 Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan 2 Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan

* Corresponding author: Jumpei Inoue

Dedicated to Professor Yoshio Yamada on the occasion of his 70th birthday

Received  January 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The second author is supported by JSPS KAKENHI Grant-in-Aid Grant Number 19K03581

Concerning a class of diffusive logistic equations, Ni [1,Abstract] proposed an optimization problem to consider the supremum of the ratio of the $L^1$ 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 $3$ in the one-dimensional case. In [1], 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 $L^1$ unbounded sequence of sub-solutions.

Citation: Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
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##### References:
 [1] Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212 [2] Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029 [3] Zaizheng Li, Zhitao Zhang. Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology. Electronic Research Archive, , () : -. doi: 10.3934/era.2021060 [4] David Aleja, Julián López-Gómez. Some paradoxical effects of the advection on a class of diffusive equations in Ecology. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3031-3056. doi: 10.3934/dcdsb.2014.19.3031 [5] Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314 [6] Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122 [7] Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943 [8] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [9] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [10] Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713 [11] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447 [12] Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067 [13] Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166 [14] Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41 [15] Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198 [16] Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232 [17] Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507 [18] Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113 [19] A. M. Micheletti, Monica Musso, A. Pistoia. Super-position of spikes for a slightly super-critical elliptic equation in $R^N$. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 747-760. doi: 10.3934/dcds.2005.12.747 [20] Alfonso Castro, Shu-Zhi Song. Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3347-3355. doi: 10.3934/dcdss.2020127

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