AIMS Mathematics, 2019, 4(3): 792-804. doi: 10.3934/math.2019.3.792.

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Existence of positive weak solutions for a nonlocal singular elliptic system

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{1,1}$ boundary,and let $s\in\left( 0,1\right) $ be such that $s<\frac{n}{2}.$ We givesufficient conditions for the existence of a weak solution $\left(u,v\right) \in H^{s}\left( \mathbb{R}^{n}\right) \times H^{s}\left(\mathbb{R}^{n}\right) $ of the nonlocal singular system $\left(-\Delta\right) ^{s}u=ad_{\Omega}^{-\gamma_{1}}v^{-\beta_{1}}$ in $\Omega,$$\left( -\Delta\right) ^{s}v=bd_{\Omega}^{-\gamma_{2}}u^{-\beta_{2}}$ in$\Omega,$ $u=v=0$ in $\mathbb{R}^{n}\setminus\Omega,$ $u>0$ in $\Omega,$ $v>0$in $\Omega,$ \ where $a$ and $b$ are nonnegative bounded measurable functionssuch that $\inf_{\Omega}a>0$ and $\inf_{\Omega}b>0.$ For the found weaksolution $\left( u,v\right) ,$ the behavior of $u$ and $v$ near$\partial\Omega$ is also investigated.
  Article Metrics

Keywords fractional singular elliptic systems; positive solutions; sub and supersolutions; Schauder fixed point theorem

Citation: Tomas Godoy. Existence of positive weak solutions for a nonlocal singular elliptic system. AIMS Mathematics, 2019, 4(3): 792-804. doi: 10.3934/math.2019.3.792


  • 1.C. Alves, Multiplicity of positive solutions for amixed boundary value problem, Rocky MT J. Math., 38 (2008), 19-39.    
  • 2.I. Bachar, H. Mâagli and V.Rădulescu, Singular solutions of a nonlinear elliptic equation ina punctured domain, Electron. J. Qual. Theo., 94 (2017), 1-19.
  • 3.B. Barrios, I. De Bonis, M. Medina, et al.Semilinear problems for the fractional laplacian with a singularnonlinearity, Open Math., 13 (2015), 390-407.
  • 4.U. Biccari, M Warma and E. Zuazua,Local elliptic regularity for the Dirichlet fractional laplacian,Adv. Nonlinear Stud., 17 (2017), 387-409.
  • 5.A. Callegari and A. Nachman, A nonlinear singularboundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl.Math., 38 (1980), 275-281.    
  • 6.M. Chhetri, P. Drabek, R. Shivaji, Analysis ofpositive solutions for classes of quasilinear singular problems on exteriordomains, Adv. Nonlinear Anal., 6 (2017), 447-459.
  • 7.M. B. Chrouda, Existence and nonexistence ofpositive solutions to the fractional equation ${\Delta ^{\frac{\alpha }{2}}}u = - {u^\gamma }$ in bounded domains, Annales Academiæ Scientiarum Fennicæ Mathematica, 42 (2017), 997-1007.    
  • 8.F. Cîrstea, M. Ghergu and V.Rădulescu, Combined effects of asymptotically linear and singularnonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math.Pure. Appl., 84 (2005), 493-508.    
  • 9.D. S. Cohen and H. B. Keller, Some positive problemssuggested by nonlinear heat generators, J. Math. Mech., 16 (1967), 1361-1376.
  • 10.M. G. Crandall, P. H. Rabinowitz and L. Tartar, Ona Dirichlet problem with a singular nonlinearity, Commun. Part. Diff.Eq., 2 (1977), 193-222.    
  • 11.L. M. Del Pezzo and A. Quaas, Globalbifurcation for fractional p-laplacian and an application, Zeitschriftfür Analysis und ihre Anwendungen, 35 (2016), 411-447.%doi: 10.4171/ZAA/1572.    
  • 12.M. A. del Pino, A global estimate for the gradientin a singular elliptic boundary value problem, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 122 (1992), 341-352.    
  • 13.J. I. Diaz, J. Hernandez and J. M. Rakotoson,On very weak positive solutions to some semilinear elliptic problemswith simultaneous singular nonlinear and spatial dependence terms, Milan J.Math., 79 (2011), 233.
  • 14.J. I. Díaz, J. M. Morel and L. Oswald, Anelliptic equation with singular nonlinearity, Commun. Part. Diff.Eq., 12 (1987), 1333-1344.    
  • 15.E. Di Nezza, G. Palatucci and E. Valdinoci,Hitchhiker's guide to the fractional Sobolev spaces, B. Sci. Math.,136 (2012), 521-573.    
  • 16.L. Dupaigne, M. Ghergu and V.Rădulescu, Lane-Emden-Fowler equations with convection andsingular potential, J. Math. Pure. Appl., 87 (2007), 563-581.    
  • 17.OK. W. Fulks and J. S. Maybee, A singularnonlinear equation, Osaka J. Math., 12 (1960), 1-19.
  • 18.A. Fiscella, R. Servadei and E.Valdinoci, Density properties for fractional Sobolev Spaces, Ann.Acad. Sci. Fenn. Math., 40 (2015), 235-253.    
  • 19.L. Gasiński and N. S. Papageorgiou,Nonlinear elliptic equations with singular terms and combinednonlinearities, Ann. Henri Poincaré, 13 (2012), 481-512.    
  • 20.M. Ghergu, V. Liskevich and Z. Sobol, Singularsolutions for second-order non-divergence type elliptic inequalities inpunctured balls, J. Anal. Math., 123 (2014), 251-279.    
  • 21.M. Ghergu, V. Rădulescu, Singularelliptic problems: bifurcation and asymptotic analysis, Oxford Lecture Seriesin Mathematics and its Applications, The Clarendon Press, OxfordUniversity Press, Oxford, 2008.
  • 22.J. Giacomoni, J. Hernandez and P. Sauvy,Quasilinear and singular elliptic systems, Adv. Nonlinear Anal.,2 (2013), 1-41.    
  • 23.J. Giacomoni, T. Mukherjee, K. Sreenadh,Positive solutions of fractional elliptic equation with critical andsingular nonlinearity, Adv. Nonlinear Anal., 6 (2017), 327-354.
  • 24.T. Godoy, A semilnear singular problem for thefractional laplacian, AIMS Mathematics, 3 (2018), 464-484.    
  • 25.A. C. Lazer and P. J. McKenna, On a singularnonlinear elliptic boundary value problem, Proc. Amer. Math. Soc.,111 (1991), 721-730.    
  • 26.H. Mâagli, Asymptotic behavior of positivesolutions of a semilinear Dirichlet problem, Nonlinear Analysis: Theory, Methods \& Applications, 74 (2011), 2941-2947.
  • 27.H. Mâagli and M. Zribi, Existence andestimates of solutions for singular nonlinear elliptic problems, J. Math.Anal. Appl., 263 (2001), 522-542.    
  • 28.G. Molica Bisci, V. Rădulescu and R. Servadei,Variational methods for nonlocal fractional problems, Encyclopedia ofMathematics and its Applications, Cambridge University Press,Cambridge, 2016.
  • 29.M. Montenegro and A. Suárez, Existenceof a positive solution for a singular system, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 140 (2010), 435-447.    
  • 30.N. S. Papageorgiou and G. Smyrlis,Nonlinear elliptic equations with singular reaction, Osaka J. Math.,53 (2016), 489-514.
  • 31.N. S. Papageorgiou, D. D. Repov and V.D. Rădulescu, Nonlinear analysis-theory and methods, SpringerMonographs in Mathematics, Springer, Cham, 2019.
  • 32.K. Ho, K. Perera, I. Sim, et al.A note on fractional p-laplacian problems with singular weights, J.Fixed Point Theory A., 19 (2017), 157-173.    
  • 33.V. D. Rădulescu, Singular phenomena innonlinear elliptic problems. From blow-up boundary solutions to equations withsingular nonlinearities, In: Handbook of Differential Equations: StationaryPartial Differential Equations (M. Chipot, Editor), North-HollandElsevier Science, Amsterdam, 4 (2007), 483-591.
  • 34.X. Ros-Oton, Nonlocal elliptic equations in boundeddomains: a survey, Publ. Mat., 60 (2016), 3-26.    
  • 35.X. Ros Oton and J. Serra, The Dirichletproblem fot the fractional laplacian: Regularity up to the boundary, J. Math.Pure. Appl., 101 (2014), 275-302.    
  • 36.R. Servadei and E. Valdinoci, Variationalmethods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst.,33 (2013), 2105-2137.
  • 37.Z. Zhang, The asymptotic behaviour of the uniquesolution for the singular Lane-Emden-Fowler equation, J. Math. Anal. Appl.,312 (2005), 33-43.    


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved