Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 13, Issue 1

Issues

Volume 13 (2015)

Semilinear problems for the fractional laplacian with a singular nonlinearity

Begoña Barrios / Ida De Bonis
  • Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza, Università di Roma, Italy
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ María Medina / Ireneo Peral
Published Online: 2015-06-11 | DOI: https://doi.org/10.1515/math-2015-0038

Abstract

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

Keywords: Fractional Laplacian; Solvability of elliptic equations; Existence and multiplicity

References

  • [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 Google Scholar

  • [2] Ambrosetti A., Rabinowitz P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 1973, 14, 349-381 CrossrefGoogle 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 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 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 CrossrefGoogle 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 Web of ScienceCrossrefGoogle Scholar

  • [7] Boccardo L., Orsina L., Semilinear elliptic equations with singular nolinearities, Calc. Var. Partial Differential Equations, 2010, 37(3-4), 363-380 Google Scholar

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

  • [9] Boccardo L., Escobedo M., Peral I., A Dirichlet problem involving critical exponent, Nonlinear Anal., 1995, 24, 1639-1848 Google Scholar

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

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

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

  • [15] Dávila J., A strong maximum principle for the Laplace equation with mixed boundary condition, J. Funct. Anal., 2001, 183, 231-244 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 Web of ScienceGoogle 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 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 CrossrefGoogle 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 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 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 Google Scholar

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

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

  • [29] Servadei R., Valdinoci E., Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 2012, 389(2), 887-898 Web of ScienceGoogle Scholar

  • [30] Servadei R., Valdinoci E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 2013, 33(5), 2105-2137 Web of ScienceGoogle 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 Web of ScienceGoogle 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 CrossrefGoogle Scholar

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

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

About the article

Received: 2014-12-16

Accepted: 2015-05-28

Published Online: 2015-06-11


Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0038.

Export Citation

©2015 Begoña Barrios et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Li Wang, Kun Cheng, and Binlin Zhang
Applied Mathematics & Optimization, 2019
[2]
R. Arora, J. Giacomoni, D. Goel, and K. Sreenadh
Asymptotic Analysis, 2019, Page 1
[7]
[8]
Adimurthi, Jacques Giacomoni, and Sanjiban Santra
Journal of Differential Equations, 2018
[9]
Annamaria Canino, Luigi Montoro, Berardino Sciunzi, and Marco Squassina
Bulletin des Sciences Mathématiques, 2017, Volume 141, Number 3, Page 223
[10]
Sarika Goyal
Complex Variables and Elliptic Equations, 2017, Volume 62, Number 2, Page 158
[11]
Boumediene Abdellaoui, María Medina, Ireneo Peral, and Ana Primo
Journal of Differential Equations, 2016, Volume 260, Number 11, Page 8160

Comments (0)

Please log in or register to comment.
Log in