Research article Special Issues

The Gelfand problem for the Infinity Laplacian

  • Received: 10 December 2021 Revised: 21 February 2022 Accepted: 22 February 2022 Published: 11 April 2022
  • We study the asymptotic behavior as p of the Gelfand problem

    {Δpu=λeuin ΩRnu=0on Ω.

    Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of

    {min{|u|Λeu,Δu}=0in Ω,u=0 on Ω.

    We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ.

    Citation: Fernando Charro, Byungjae Son, Peiyong Wang. The Gelfand problem for the Infinity Laplacian[J]. Mathematics in Engineering, 2023, 5(2): 1-28. doi: 10.3934/mine.2023022

    Related Papers:

    [1] Chentong Li, Jinyan Wang, Jinhu Xu, Yao Rong . The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments. Mathematical Biosciences and Engineering, 2022, 19(5): 4690-4702. doi: 10.3934/mbe.2022218
    [2] Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva . An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences and Engineering, 2010, 7(1): 123-147. doi: 10.3934/mbe.2010.7.123
    [3] Matthew D. Johnston, Bruce Pell, David A. Rubel . A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity. Mathematical Biosciences and Engineering, 2023, 20(9): 16083-16113. doi: 10.3934/mbe.2023718
    [4] Ali Mai, Guowei Sun, Lin Wang . The impacts of dispersal on the competition outcome of multi-patch competition models. Mathematical Biosciences and Engineering, 2019, 16(4): 2697-2716. doi: 10.3934/mbe.2019134
    [5] Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin . Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068
    [6] Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis . A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme. Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679
    [7] Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283
    [8] Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135
    [9] Yanxia Dang, Zhipeng Qiu, Xuezhi Li . Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences and Engineering, 2017, 14(4): 901-931. doi: 10.3934/mbe.2017048
    [10] Azmy S. Ackleh, Shuhua Hu . Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences and Engineering, 2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133
  • We study the asymptotic behavior as p of the Gelfand problem

    {Δpu=λeuin ΩRnu=0on Ω.

    Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of

    {min{|u|Λeu,Δu}=0in Ω,u=0 on Ω.

    We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ.





    [1] B. Abdellaoui, I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Ann. Mat. Pura Appl. IV. Ser., 182 (2003), 247–270. https://doi.org/10.1007/s10231-002-0064-y doi: 10.1007/s10231-002-0064-y
    [2] A. Anane, J. L. Lions, Simplicitè et isolation de la premiere valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725–728.
    [3] T. Bhattacharya, E. DiBenedetto, J. Manfredi, Limit as p of Δpup=f and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, 1989, 15–68.
    [4] M. Bocea, M. Mihǎilescu, Existence of nonnegative viscosity solutions for a class of problems involving the -Laplacian, Nonlinear Differ. Equ. Appl., 23 (2016), 11. https://doi.org/10.1007/s00030-016-0373-2 doi: 10.1007/s00030-016-0373-2
    [5] G. Bratu, Sur les équations intégrales non linéaires, Bulletin de la S. M. F., 42 (1914), 113–142.
    [6] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. Theor., 10 (1986), 55–64. https://doi.org/10.1016/0362-546X(86)90011-8 doi: 10.1016/0362-546X(86)90011-8
    [7] X. Cabré, M. Sanchón, Semi-stable and extremal solutions of reaction equations involving the p-Laplacian, Commun. Pure Appl. Anal., 6 (2007), 43–67. https://doi.org/10.3934/cpaa.2007.6.43 doi: 10.3934/cpaa.2007.6.43
    [8] S. Chandrasekhar, An introduction to the study of stellar structures, New York: Dover, 1957.
    [9] S. Chanillo, M. Kiessling, Surfaces with prescribed Gauss curvature, Duke Math. J., 105 (2000), 309–353. https://doi.org/10.1215/S0012-7094-00-10525-X doi: 10.1215/S0012-7094-00-10525-X
    [10] F. Charro, E. Parini, Limits as p of p-laplacian problems with a superdiffusive power-type nonlinearity: positive and sign-changing solutions, J. Math. Anal. Appl., 372 (2010), 629–644. https://doi.org/10.1016/j.jmaa.2010.07.005 doi: 10.1016/j.jmaa.2010.07.005
    [11] F. Charro, E. Parini, Limits as p of p-laplacian eigenvalue problems perturbed with a concave or convex term, Calc. Var., 46 (2013), 403–425. https://doi.org/10.1007/s00526-011-0487-7 doi: 10.1007/s00526-011-0487-7
    [12] F. Charro, I. Peral, Limit branch of solutions as p for a family of sub-diffusive problems related to the p-laplacian, Commun. Part. Diff. Eq., 32 (2007), 1965–1981. https://doi.org/10.1080/03605300701454792 doi: 10.1080/03605300701454792
    [13] F. Charro, I. Peral, Limits as p of p-Laplacian concave-convex problems, Nonlinear Anal. Theor., 75 (2012), 2637–2659. https://doi.org/10.1016/j.na.2011.11.009 doi: 10.1016/j.na.2011.11.009
    [14] M. G. Crandall, H. Ishii, P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1–67. https://doi.org/10.1090/S0273-0979-1992-00266-5 doi: 10.1090/S0273-0979-1992-00266-5
    [15] J. Dávila, Singular solutions of semi-linear elliptic problems, In: Handbook of differential equations: stationary partial differential equations, Amsterdam: Elsevier/North-Holland, 2008, 83–176. https://doi.org/10.1016/S1874-5733(08)80019-8
    [16] M. Del Pino, J. Dolbeault, M. Musso, Multiple bubbling for the exponential nonlinearity in the slightly supercritical case, Commun. Pure Appl. Anal., 5 (2006), 463–482. https://doi.org/10.3934/cpaa.2006.5.463 doi: 10.3934/cpaa.2006.5.463
    [17] R. Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme (German), Germany: B. G. Teubner, 1907.
    [18] N. Fukagai, M. Ito, K. Narukawa, Limit as p of p-Laplace eigenvalue problems and L-inequality of the Poincare type, Differ. Integral Equ., 12 (1999), 183–206.
    [19] J. P. García Azorero, I. Peral, Existence and nonuniqueness for the p-laplacian: nonlinear eigenvalues, Commun. Part. Diff. Eq., 12 (1987), 1389–1430.
    [20] J. García Azorero, I. Peral Alonso, On a Emden-Fowler type equation, Nonlinear Anal. Theor., 18 (1992), 1085–1097. https://doi.org/10.1016/0362-546X(92)90197-M doi: 10.1016/0362-546X(92)90197-M
    [21] J. García Azorero, I. Peral Alonso, J. P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal. Theor., 22 (1994), 481–498. https://doi.org/10.1016/0362-546X(94)90169-4 doi: 10.1016/0362-546X(94)90169-4
    [22] I. M. Gel'fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Trans., 29 (1963), 295–381.
    [23] J.-F. Grosjean, p-Laplace operator and diameter of manifolds, Ann. Glob. Anal. Geom., 28 (2005), 257–270. https://doi.org/10.1007/s10455-005-6637-4 doi: 10.1007/s10455-005-6637-4
    [24] J. Jacobsen, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differ. Equations, 184 (2002), 283–298. https://doi.org/10.1006/jdeq.2001.4151 doi: 10.1006/jdeq.2001.4151
    [25] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51–74. https://doi.org/10.1007/BF00386368 doi: 10.1007/BF00386368
    [26] D. D. Joseph, T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241–269. https://doi.org/10.1007/BF00250508 doi: 10.1007/BF00250508
    [27] P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions, Helsinki: Suomalainen Tiedeakatemia, 1998.
    [28] P. Juutinen, Principal eigenvalue of a badly degenerate operator, J. Differ. Equations, 236 (2007), 532–550. https://doi.org/10.1016/j.jde.2007.01.020 doi: 10.1016/j.jde.2007.01.020
    [29] P. Juutinen, P. Lindqvist, On the higher eigenvalues for the -eigenvalue problem, Calc. Var., 23 (2005), 169–192. https://doi.org/10.1007/s00526-004-0295-4 doi: 10.1007/s00526-004-0295-4
    [30] P. Juutinen, P. Lindqvist, J. Manfredi, The -eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89–105. https://doi.org/10.1007/s002050050157 doi: 10.1007/s002050050157
    [31] B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1–22.
    [32] P. Lindqvist, On the equation div(|u|p2u)+λ|u|p2u=0, Proc. Amer. Math. Soc., 109 (1990), 157–164. https://doi.org/10.1090/S0002-9939-1990-1007505-7 doi: 10.1090/S0002-9939-1990-1007505-7
    [33] P. Lindqvist, Addendum: "On the equation div(|u|p2u)+λ|u|p2u=0'' [Proc. Amer. Math. Soc. 109 (1990), no. 1,157–164; MR1007505 (90h: 35088)], Proc. Amer. Math. Soc., 116 (1992), 583–584. https://doi.org/10.1090/S0002-9939-1992-1139483-6
    [34] P. Lindqvist, J. Manfredi, The Harnack inequality for -harmonic functions, Electron. J. Diff. Eqns., 5 (1995), 1–5.
    [35] P. Lindqvist, J. Manfredi, Note on -superharmonic functions, Revista Matemática de la Universidad Complutense de Madrid, 10 (1997), 471–480.
    [36] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441–467. https://doi.org/10.1137/1024101 doi: 10.1137/1024101
    [37] J. Liouville, Sur l'équation aux différences partielles d2logλdudv±λ2a2=0, Journal de mathématiques pures et appliquées 1re série, 18 (1853), 71–72.
    [38] M. Mihǎilescu, D. Stancu-Dumitru, C. Varga, The convergence of nonnegative solutions for the family of problems Δpu=λeu as p, ESAIM: COCV, 24 (2018), 569–578. https://doi.org/10.1051/cocv/2017048 doi: 10.1051/cocv/2017048
    [39] I. Peral, Some results on quasilinear elliptic equations: growth versus shape, In: Proceedings of the second school of nonlinear functional analysis and applications to differential equations, Trieste: World Scientific, 1998,153–202.
    [40] O. W. Richardson, The emission of electricity from hot Bodies, India: Longmans, Green and Company, 1921.
    [41] M. Sanchón, Regularity of the extremal solution of some nonlinear elliptic problems involving the p-Laplacian, Potential Anal., 27 (2007), 217–224. https://doi.org/10.1007/s11118-007-9053-5 doi: 10.1007/s11118-007-9053-5
    [42] G. W. Walker, Some problems illustrating the forms of nebulae, Proc. R. Soc. Lond. A, 91 (1915), 410–420. https://doi.org/10.1098/rspa.1915.0032 doi: 10.1098/rspa.1915.0032
    [43] Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56 (2007), 947–964.
  • This article has been cited by:

    1. Yixiang Wu, Necibe Tuncer, Maia Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, 2017, 22, 1553-524X, 1167, 10.3934/dcdsb.2017057
    2. Junping Shi, Yixiang Wu, Xingfu Zou, Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction–Diffusion Chemostat Model, 2020, 32, 1040-7294, 1085, 10.1007/s10884-019-09763-0
    3. Yixiang Wu, Xingfu Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, 2016, 261, 00220396, 4424, 10.1016/j.jde.2016.06.028
    4. Lin Zhao, Zhi-Cheng Wang, Shigui Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, 2020, 51, 14681218, 102966, 10.1016/j.nonrwa.2019.102966
    5. Jing Ge, Ling Lin, Lai Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, 2017, 22, 1553-524X, 2763, 10.3934/dcdsb.2017134
    6. Yuan Lou, Rachidi B. Salako, Control Strategies for a Multi-strain Epidemic Model, 2022, 84, 0092-8240, 10.1007/s11538-021-00957-6
    7. Jinsheng Guo, Shuang-Ming Wang, Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay, 2022, 7, 2473-6988, 6331, 10.3934/math.2022352
    8. Rachidi B. Salako, Impact of population size and movement on the persistence of a two-strain infectious disease, 2023, 86, 0303-6812, 10.1007/s00285-022-01842-z
    9. Yuan Lou, Rachidi B. Salako, Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases, 2023, 370, 00220396, 424, 10.1016/j.jde.2023.06.018
    10. Jonas T. Doumatè, Tahir B. Issa, Rachidi B. Salako, Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023213
    11. Azmy S. Ackleh, Nicolas Saintier, Aijun Zhang, A multiple-strain pathogen model with diffusion on the space of Radon measures, 2025, 140, 10075704, 108402, 10.1016/j.cnsns.2024.108402
    12. Jamal Adetola, Keoni G. Castellano, Rachidi B. Salako, Dynamics of classical solutions of a multi-strain diffusive epidemic model with mass-action transmission mechanism, 2025, 90, 0303-6812, 10.1007/s00285-024-02167-9
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2463) PDF downloads(229) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog