Abstract
We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abdellaoui B, Boucherif A, Touaoula T M. Fractional parabolic problems with a nonlocal initial condition. Moroccan J Pure Appl Anal, 2017, 3(1): 116–132
Abdellaoui B, Medina M, Peral I, Primo A. The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian. J Differential Equations, 2016, 260(11): 8160–8206
Abdellaoui B, Medina M, Peral I, Primo A. Optimal results for the fractional heat equation involving the Hardy potential. Nonlinear Anal, 2016, 140: 166–207
Adimurthi A, Giacomoni J, Santra S. Positive solutions to a fractional equation with singular nonlinearity. J Differential Equations, 2018, 265(4): 1191–1226
Alibaud N, Andreianov B, Bendahmane M. Renormalized solutions of the fractional Laplace equation. C R Math Acad Sci Paris, 2010, 348(13/14): 759–762
Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, 116. 2nd ed. Cambridge: Cambridge Univ Press, 2009
Barrios B, De Bonis I, Medina M, Peral I. Semilinear problems for the fractional laplacian with a singular nonlinearity. Open Math, 2015, 13: 390–407
Barrios B, Medina M, Peral I. Some remarks on the solvability of non-local elliptic problems with the Hardy potential. Commun Contemp Math, 2014, 16(4): 1350046, 29
Bénilan P, Boccardo L, Gallouet T, Gariepy R, Pierre M, Vasquez J L. An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annali Scuola Norm Sup Pisa, 1995, 22(4): 241–273
Bisci G M, Radulescu V D, Servadei R. Variational methods for nonlocal fractional problems//Encyclopedia of Mathematics and its Applications, Vol 162. Cambridge: Cambridge University Press, 2016
Boccardo L, Orsina L. Semilinear elliptic equations with singular nonlinearities. Calc Var Partial Differential Equations, 2010, 37(3/4): 363–380
Canino A, Montoro L, Sciunzi B, Squassina M. Nonlocal problems with singular nonlinearity. Bull Sci Math, 2017, 141(3): 223–250
Crandall M G, Rabinowitz P H, Tartar L. On a dirichlet problem with a singular nonlinearity. Comm Partial Differential Equations, 1977, 2(2): 193–222
Danielli D, Salsa S. Obstacle problems involving the fractional Laplacian//Recent Developments in Nonlocal Theory. Berlin: De Gruyter, 2018: 81–164
De Cave L M, Oliva F. Elliptic equations with general singular lower order term and measure data. Nonlinear Anal, 2015, 128: 391–411
Demengel F, Demengel G. Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext. London: Springer; Les Ulis: EDP Sciences, 2012
Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136(5): 521–573
Dipierro, S, Figalli A, Valdinoci E. Strongly nonlocal dislocation dynamics in crystals. Comm Partial Differential Equations, 2014, 39(12): 2351–2387
Fiscella A, Servadei R Valdinoci E. Density properties for fractional Sobolev spaces. Ann Acad Sci Fenn Math, 2015, 40(1): 235–253
Kenneth K H, Petitta F, Ulusoy S. A duality approach to the fractional Laplacian with measure data. Publ Mat, 2011, 55(1): 151–161
Klimsiak T. Reduced measures for semilinear elliptic equations involving Dirichlet operators. Nonlinear Anal, 2016, 55(4): Art 78, 27
Kufner A, John O, Fučík S. Function Spaces. Leyden, Academia, Prague: Noordhoff International Publishing, 1977
Landkof N. Foundations of Modern Potential Theory. Die Grundlehren der Mathematischen Wissenschaften, Vol 180. New York, Heidelberg: Springer-Verlag, 1972
Lazer A C, McKenna P J. On a singular nonlinear elliptic boundary-value problem. Proc Amer Math Soc, 1991, 111(3): 721–730
Leonori T, Peral I, Primo A, Soria F. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Contin Dyn Syst, 2015, 35(12): 6031–6068
Oliva F, Petitta F. On singular elliptic equations with measure sources. ESAIM Control Optim Calc Var, 2016, 22(1): 289–308
Oliva F, Petitta F. Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J Differential Equations, 2018, 264(1): 311–340
Petitta F. Some remarks on the duality method for integro-differential equations with measure data. Adv Nonlinear Stud, 2016, 16(1): 115–124
Ponce A C. Elliptic PDEs, Measures and Capacities. EMS Tracts in Mathematics, 23. Zürich: European Mathematical Society (EMS), 2016
Sire Y, Valdinoci E. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J Funct Anal, 2009, 256(6): 1842–1864
Stuart C A. Existence and approximation of solutions of non-linear elliptic equations. Math Z, 1976, 147(1): 53–63
Sun Y J, Zhang D Z. The role of the power 3 for elliptic equations with negative exponents. Calc Var Partial Differential Equations, 2014, 49(3/4): 909–922
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Youssfi, A., Ould Mohamed Mahmoud, G. On Singular Equations Involving Fractional Laplacian. Acta Math Sci 40, 1289–1315 (2020). https://doi.org/10.1007/s10473-020-0509-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0509-7