Open Access Journals

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

Export file:

Format

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

Content

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

Received: , Accepted: , Published:

Abstract Full Text(HTML) Figure/Table Related pages

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.

Figure/Table

Supplementary

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

References:

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.

show all references

Reader Comments

your name: * your email: *

Download full text in PDF

Associated material

Metrics