Large solutions for cooperative logistic systems: existence and uniqueness in star-shaped domains

[1]P. Álvarez-Caudevilla and J. López-Gómez, Metasolutions for cooperative systems, Nonl. Anal. RWA 9 (2008), 1119—1157. 10.1016/j.nonrwa.2007.02.010Álvarez-CaudevillaP.López-GómezJ.Metasolutions for cooperative systemsNonl. Anal. RWA920081119115710.1016/j.nonrwa.2007.02.010Open DOISearch in Google Scholar

[2]H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620—709.AmannH.Fixed point equations and nonlinear eigenvalue problems in ordered Banach spacesSIAM Rev18197662070910.1137/1018114Search in Google Scholar

[3]S. Cano-Casanova and J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl. 352 (2009), 166—174. 10.1016/j.jmaa.2008.06.022Cano-CasanovaS.López-GómezJ.Blow-up rates of radially symmetric large solutionsJ. Math. Anal. Appl3522009166174doi 10.1016/j.jmaa.2008.06.022Open DOISearch in Google Scholar

[4]S. Dumont, L. Dupaigne, O. Goubet and V. Radulescu, Back to the Keller—Osserman condition for boundary blow-up solutions, Adv. Nonl. Studies 7 (2007), 271—298.DumontS.DupaigneL.GoubetO.RadulescuV.Back to the Keller—Osserman condition for boundary blow-up solutionsAdv. Nonl. Studies7200727129810.1515/ans-2007-0205Search in Google Scholar

[5]C. Faber, Beweis das unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisdörmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math. Phys. (1923), 169—171.FaberC.Beweis das unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisdörmige den tiefsten Grundton gibtSitzungsber. Bayer. Akad. der Wiss. Math. Phys1923169171Search in Google Scholar

[6]J. Garcia-Melián, J. Rossi and J. C. Sabina de Lis, Elliptic systems with boundary blow-up: Existence, uniqueness and applications to removability of singularities, Commun. Pure Appl. Anal. 15 (2) (2016) 549—562. 10.3934/cpaa.2016.15.549Garcia-MeliánJ.RossiJ.Sabina de LisJ. C.Elliptic systems with boundary blow-up: Existence, uniqueness and applications to removability of singularitiesCommun. Pure Appl. Anal1522016549562doi 10.3934/cpaa.2016.15.549Open DOISearch in Google Scholar

[7]J. Garcia-Melián and A. Suárez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Adv. Nonlinear Stud. 3 (2003), no. 2, 193-206. 10.1515/ans-2003-0203Garcia-MeliánJ.SuárezA.Existence and uniqueness of positive large solutions to some cooperative elliptic systemsAdv. Nonlinear Stud32003219320610.1515/ans-2003-0203Open DOISearch in Google Scholar

[8]J. B. Keller, On solutions of Au = f (u), Comm. Pure and Appl. Maths., X (1957), 503—510.KellerJ. B.On solutions of Au = f (u)Comm. Pure and Appl. Maths., X195750351010.1002/cpa.3160100402Search in Google Scholar

[9]E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 91 (1925), 97—100.KrahnE.Über eine von Rayleigh formulierte Minimaleigenschaft des KreisesMath. Ann9119259710010.1007/BF01208645Search in Google Scholar

[10]J. López-Gómez, Uniqueness of radially symmetric large solutions, Disc. Cont. Dynam. Systems, Proceedings of the sixth AIMS Conference, Poitiers, Supplement 2007, 677—686.López-GómezJ.Uniqueness of radially symmetric large solutionsDisc. Cont. Dynam. Systems, Proceedings of the sixth AIMS Conference, Poitiers, Supplement2007677686Search in Google Scholar

[11]J. López-Gómez, Metasolutions of Parabolic Problems in Population Dynamics, CRC Press, Boca Raton, 2015.López-GómezJ.Metasolutions of Parabolic Problems in Population DynamicsCRC PressBoca Raton201510.1201/b19418Search in Google Scholar

[12]J. López-Gómez and L. Maire, Uniqueness of large positive solutions for a class of radially symmetric cooperative systems, J. Math. Anal. Appl. 435 (2016), 1738—1752. 10.1016/j.jmaa.2015.11.026López-GómezJ.MaireL.Uniqueness of large positive solutions for a class of radially symmetric cooperative systemsJ. Math. Anal. Appl43520161738175210.1016/j.jmaa.2015.11.026Open DOISearch in Google Scholar

[13]J. López-Gómez and L. Maire, Boundary blow-up rates and uniqueness of the large solution for cooperative elliptic systems of logistic type, Nonl. Anal. RWA 33 (2017) 298—316. 10.1016/j.nonrwa.2016.07.001López-GómezJ.MaireL.Boundary blow-up rates and uniqueness of the large solution for cooperative elliptic systems of logistic typeNonl. Anal. RWA33201729831610.1016/j.nonrwa.2016.07.001Open DOISearch in Google Scholar

[14]J. López-Gómez and L. Maire, Coupled versus uncoupled blow-up rates on cooperative n-species logistic systems, Advanced Nonlinear Studies, (2017) 10.1515/ans-2016-6018López-GómezJ.MaireL.Coupled versus uncoupled blow-up rates on cooperative n-species logistic systemsAdvanced Nonlinear Studies2017doi 10.1515/ans-2016-6018Open DOISearch in Google Scholar

[15]J. López-Gómez and L. Maire, Uniqueness of large positive solution, in pressLópez-GómezJ.MaireL.Uniqueness of large positive solution, in pressSearch in Google Scholar

[16]J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns. 7 (1994), 383—398.López-GómezJ.Molina-MeyerM.The maximum principle for cooperative weakly coupled elliptic systems and some applicationsDiff. Int. Eqns71994383398Search in Google Scholar

[17]M. Molina-Meyer, Existence and uniqueness of coexistence states for some nonlinear elliptic systems, Nonl. Anal. TMA 25 (1995), 279—296.Molina-MeyerM.Existence and uniqueness of coexistence states for some nonlinear elliptic systemsNonl. Anal. TMA25199527929610.1016/0362-546X(94)00137-7Search in Google Scholar

[18]M. Molina-Meyer, Global attractivity and singular perturbation for a class of nonlinear cooperative systems, J. Diff. Eqns. 128 (1996), 347—378.Molina-MeyerM.Global attractivity and singular perturbation for a class of nonlinear cooperative systemsJ. Diff. Eqns128199634737810.1006/jdeq.1996.0099Search in Google Scholar

[19]M. Molina-Meyer, Uniqueness and existence of positive solutions for weakly coupled general sublinear systems, Nonl. Anal. TMA 30 (1997), 5375—5380.Molina-MeyerM.Uniqueness and existence of positive solutions for weakly coupled general sublinear systemsNonl. Anal. TMA3019975375538010.1016/S0362-546X(97)00380-5Search in Google Scholar

[20]R. Osserman, On the inequality Au > f (u), Pacific J. of Maths., 7 (1957), 1641—1647.OssermanR.On the inequality Au > f (u)Pacific J. of Maths719571641164710.2140/pjm.1957.7.1641Search in Google Scholar