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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access June 11, 2015

Semilinear problems for the fractional laplacian with a singular nonlinearity

  • Begoña Barrios , Ida De Bonis , María Medina and Ireneo Peral
From the journal Open Mathematics


The aim of this paper is to study the solvability of the problem

where Ω is a bounded smooth domain of RN, N > 2s, M ε {0, 1}, 0 < s < 1, γ > 0, λ > 0, p > 1 and f is a nonnegative function. We distinguish two cases:

– For M = 0, we prove the existence of a solution for every γ > 0 and λ > 0.

A1 – For M = 1, we consider f ≡ 1 and we find a threshold ʌ such that there exists a solution for every 0 < λ < ʌ ƒ, and there does not for λ > ʌ ƒ


[1] Ambrosetti A., Brezis H., Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 1994, 122(2), 519-543 10.1006/jfan.1994.1078Search in Google Scholar

[2] Ambrosetti A., Rabinowitz P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 1973, 14, 349-381 10.1016/0022-1236(73)90051-7Search in Google Scholar

[3] Arcoya D., Boccardo L., Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities, Differential Integral Equations, 2013, 26, 119-128 Search in Google Scholar

[4] Arcoya D., Moreno-Merida L., Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity, Nonlinear Anal., 2014, 95, 281-291 10.1016/ in Google Scholar

[5] Barrios B., Colorado E., Servadei R., Soria F., A critical fractional equation with concave-convex nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire (in press), DOI: 10.1016/j.anihpc.2014.04.003 10.1016/j.anihpc.2014.04.003Search in Google Scholar

[6] 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 10.1142/S0219199713500466Search in Google Scholar

[7] Boccardo L., Orsina L., Semilinear elliptic equations with singular nolinearities, Calc. Var. Partial Differential Equations, 2010, 37(3-4), 363-380 10.1007/s00526-009-0266-xSearch in Google Scholar

[8] Boccardo L., A Dirichlet problem with singular and supercritical nonlinearities, Nonlinear Anal., 2012, 75, 4436-4440 10.1016/ in Google Scholar

[9] Boccardo L., Escobedo M., Peral I., A Dirichlet problem involving critical exponent, Nonlinear Anal., 1995, 24, 1639-1848 10.1016/0362-546X(94)E0054-KSearch in Google Scholar

[10] Brezis H. , Kamin S., Sublinear elliptic equations in Rn, Manuscripta Math., 1992, 74, 87–106 10.1007/BF02567660Search in Google Scholar

[11] Brezis H., Nirenberg L., H1 versus C1 local minimizers, C. R. Acad. Sci. Paris t., 1993, 317, 465-472 Search in Google Scholar

[12] Canino A., Degiovanni M., A variational approach to a class of singular semilinear elliptic equations, Journal of Convex Analysis, 2004, 11(1), 147-162 Search in Google Scholar

[13] Crandall M. G., Rabinowitz P. H., Tartar L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 1977, 2, 193-222 10.1080/03605307708820029Search in Google Scholar

[14] Coclite M. M., Palmieri G., On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 1989, 14(10), 1315-1327 10.1080/03605308908820656Search in Google Scholar

[15] Dávila J., A strong maximum principle for the Laplace equation with mixed boundary condition, J. Funct. Anal., 2001, 183, 231-244 10.1006/jfan.2000.3729Search in Google Scholar

[16] Di Nezza E., Palatucci G., Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 2012, 136(5), 521-573 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[17] García Azorero J. P., Peral I., Multiplicity of solutions for elliptic problems with critical exponents or with a non-symmetric term, Transactions American Mathematical Society, 1991, 323(2), 877-895 10.1090/S0002-9947-1991-1083144-2Search in Google Scholar

[18] García Azorero J. P., Manfredi J. J., Peral I., Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2000, 2(3), 385-404 10.1142/S0219199700000190Search in Google Scholar

[19] Ghergu M., Radulescu V., Singular elliptic problems with convection term in anisotropic media, Mathematical analysis and applications, 2006, 74-89, AIP Conf. Proc., 835, Amer. Inst. Phys., Melville, NY 10.1063/1.2205038Search in Google Scholar

[20] Ghoussoub N., Preiss D., A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1989, 6(5), 321-330 10.1016/s0294-1449(16)30313-4Search in Google Scholar

[21] Hirano N., Saccon C., Shioji N., Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 2008, 245, 1997-2037 10.1016/j.jde.2008.06.020Search in Google Scholar

[22] Greco A., Servadei R., Hopf’s lemma and constrained radial symmetry for the fractional laplacian, preprint Search in Google Scholar

[23] Lair A. V., Shaker A. W., Classical and Weak Solutions of a Singular Semilinear Elliptic Problem, Journal of Mathematical Analysis and Applications, 1997, 211, 371-385 10.1006/jmaa.1997.5470Search in Google Scholar

[24] Landkof N., Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer- Verlag, New York-Heidelberg, 1972 Search in Google Scholar

[25] Lazer A. C., McKenna P. J., On a Singular Nonlinear Elliptic Boundary-Value Problem, Proceedings of the American Mathematical Society, 1991, 111(3), 721-730 10.1090/S0002-9939-1991-1037213-9Search in Google Scholar

[26] Lazer A. C., McKenna P. J., On Singular Boundary Value Problems for the Monge-Ampère Operator, Journal of Mathematical Analysis and Applications, 1996, 197, 341-362 10.1006/jmaa.1996.0024Search in Google Scholar

[27] Leonori T., Peral I., Primo A., Soria F., Basic estimates for solution of elliptic and parabolic equations for a class of nonlocal operators, preprint Search in Google Scholar

[28] Ros-Oton X., Serra J., The Dirichlet problem for the fractional laplacian: regularity up to the boundary, J. Math. Pures Appl. (9), 2014, 101(3), 275-302 10.1016/j.matpur.2013.06.003Search in Google Scholar

[29] Servadei R., Valdinoci E., Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 2012, 389(2), 887-898 10.1016/j.jmaa.2011.12.032Search in Google Scholar

[30] Servadei R., Valdinoci E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 2013, 33(5), 2105-2137 10.3934/dcds.2013.33.2105Search in Google Scholar

[31] Silvestre L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 2007, 60(1), 67-112 10.1002/cpa.20153Search in Google Scholar

[32] Stampacchia G., Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier (Grenoble), 1965, 15, fasc. 1, 189-258 10.5802/aif.204Search in Google Scholar

[33] Stein E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 Search in Google Scholar

[34] Stuart C. A., Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rational Mech. Anal., 1991, 113, 65-96 10.1007/BF00380816Search in Google Scholar

[35] Zhang Z., Boundary behavior of solutions to some singular elliptic boundary value problems. Nonlinear Anal., 2008, 69(7), 2293-2302 10.1016/ in Google Scholar

Received: 2014-12-16
Accepted: 2015-5-28
Published Online: 2015-6-11

©2015 Begoña Barrios et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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