Abstract
In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent
Similar content being viewed by others
References
D. Averna, S. Tersian, E. Tornatore, on the existence and multiplicity of solutions for Dirichlet’s problem for fractional equations. Fract. Calc. Appl. Anal. 19, No 1 (2016), 253–266; DOI:10.1515/fca-2016-0014 https://www.degruyter.com/view/journals/fca/19/1/fca.19.issue-1.xml.
B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave-convex power nonlinearities. Ann. I.H. Poincaré 32, No 4 (2015), 875–900.
B. Barrios, I. De Bonis, M. Medina, I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math. 13, (2015), 390–407.
L. Brasco, G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. 37, (2014), 769–799.
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Diff. Equations 32, (2007), 1245–1260.
W. Chen, S. Mosconi, M. Squassina, Nonlocal problems with critical Hardy non-linearity. J. of Funct. Anal. 275, No 11 (2018), 3065–3114.
G. Devillanova, G. Carlo Marano, A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19, No 2 (2016), 319–356; DOI:10.1515/fca-2016-0018 https://www.degruyter.com/view/journals/fca/19/2/fca.19.issue-2.xml.
Y. Fang, Existence uniqueness of positive solution to a fractional laplacians with singular non linearity. Preprint (2014), http://arxiv.org/pdf/1403.3149.pdf.
A. Ghanmi, K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fractional Differential Calculus 6, No 2 (2016), 201–217.
A. Ghanmi, K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator. Complex Variables and Elliptic Equations 61, No 9 (2016), 1199–1216.
N. Ghoussoub, D. Preiss, A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, No 5 (1989), 321–330.
G. Molica Bisci, V. Radulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Math. and its Appl. 162, Cambridge University Press Cambridge, (2016).
T. T. Mukherjee, K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities. Electr. J. of Differential Equations 2016, No 54 (2016), 1–23.
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhicker’s guide to the fractional Sobolev space. Bull. Sci. Math. 136, (2012), 521–537.
P. P. Tankov, R. Cont, Financial Modelling with Jump Processes Chapman and Hall, CRC Financial Mathematics Series Boca Raton, (2004).
E. Valdinoci, From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. 49, (2009), 33–44.
K. Saoudi, A critical fractional elliptic equation with singular nonlinearities. Fract. Calc. Appl. Anal. 20, No 6 (2017), 1507–153010.1515/fca-2017-0079 https://www.degruyter.com/view/journals/fca/20/6/fca.20.issue-6.xml.
K. Saoudi, S. Ghosh, D. Choudhuri, Multiplicity and Höölder regularity of solutions for a nonlocal elliptic PDE involving singularity. J. of Math. Physics 60, No 10 (2019), # 101509.
R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for nonlocal critical equations in low dimension. Commun. Pure Appl. Anal. 12, No 6 (2013), 2445–2464.
R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems 33, No 5 (2013), 2105–2137.
R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators. J. Math. Anal. and Appl. 389, No 2 (2012), 887–898.
R L. Schilling, R. Song, Z. Vondracek, Bernstein Functions. Theory and Applications Gruyter Studies in Mathematics 37, Walter de Gruyter and Co., Berlin (2012).
Q.M. Zhou, K.Q. Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian. Fract. Calc. Appl. Anal. 18, No 1 (2015), 133–14510.1515/fca-2015-0009 https://www.degruyter.com/view/journals/fca/18/1/fca.18.issue-1.xml.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Daoues, A., Hammami, A. & Saoudi, K. Multiple Positive Solutions for a Nonlocal Pde with Critical Sobolev-Hardy and Singular Nonlinearities Via Perturbation Method.. Fract Calc Appl Anal 23, 837–860 (2020). https://doi.org/10.1515/fca-2020-0042
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2020-0042