Abstract
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
Here, \(\Omega \) is a bounded domain of \(\mathbb {R}^N\), \(s\in (0,1)\), \(\alpha ,\lambda \) and \(\beta \) are positive real parameters, \(N>2s\), \(\gamma \in (0,1)\), \(0<b<\min \{N,4s\}\), \(2_b^*=\frac{2N-b}{N-2s}\) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and \(\mu \) is a bounded Radon measure in \(\Omega \).
Similar content being viewed by others
References
Abdellaoui, B., Bentifour, R.: Caffarelli–Kohn–Nirenberg type inequalities of fractional order and applications. J. Funct. Anal. 272, 3998–4029 (2017)
Adimurthi, Giacomoni J: Multiplicity of positive solutions for a singular and critical elliptic problem in \({\mathbb{R}}^2\). Commun. Contemp. Math. 8(5), 621–656 (2006)
Applebaum, D.: Lévy processes-from probability to finance and quantum groups. Notices Am. Math. Soc. 51(11), 1336–1347 (2004)
Barrios, B., Medina, M., Peral, I.: Some remarks on the solvability of nonlocal elliptic problems with the Hardy potential, Commun. Contemp. Math., 16, 1350046, 29 pp, (2014)
Boccardo, L., Gallouet, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations involving measure data. Ann. Inst. H. Poincaré Anal. Non Lineaire 13, 539–551 (1996)
Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. 37, 363–380 (2010)
Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)
Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–90 (1983)
Buffoni, B., Jeanjean, L., Stuart, C.A.: Existence of a nontrivial solution to a strongly indefinite semilinear equation. Proc. Am. Math. Soc. 119(1), 179–186 (1993)
Canino, A., Montoro, L., Sciunzi, B., Squassina, M.: Nonlocal problems with singular nonlinearity, Bull. Sci. math. (2017)
Chen, Y.H., Liu, C.: Ground state solutions for non-autonomous fractional Choquard equations. Nonlinearity 29, 1827–1842 (2016)
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(3), 837–860 (2020)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Fall, M. M.: Semilinear elliptic equations for the fractional Laplacian with Hardy potential, arXiv:1109.5530v4 [math.AP] (24 Oct 2012)
Fiscella, A., Pucci, P.: On certain nonlocal Hardy–Sobolev critical elliptic Dirichlet problems. Adv. Differ. Equ. 21(5–6), 571–599 (2016)
Gao, F., Yang, M.: On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. Sci. China Math. 61, 1219–1242 (2018)
Ghanmi, A., Saoudi, K.: The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fract. Differ. Calc. 6(2), 201–217 (2016)
Ghosh, S., Choudhuri, D., Giri, R.K.: Singular nonlocal problem involving measure data. Bull. Braz. Math. Soc. 50(1), 187–209 (2018)
Giacomoni, J., Mukherjee, T., Sreenadh, K.: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6(3), 327–354 (2017)
Giacomoni, J., Mukherjee, T., Sreenadh, K.: Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity. J. Math. Anal. Appl. 467, 638–672 (2018)
Giacomoni, J., Goel, D., Sreenadh, K.: Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities, arXiv:2002.02937v1 [math.AP] (7 Feb 2020)
Haitao, Y.: Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differ. Equ. 189, 487–512 (2003)
Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337, 1317–1368 (2015)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)
Lieb, E., Loss, M.:“Analysis”, Graduate Studies in Mathematics, AMS, Providence, Rhode island, (2001)
Lü, D., Xu, G.: On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity. Appl. Anal. 97(2), 255–273 (2018)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Panda, A., Ghosh, S., Choudhuri, D.: Elliptic partial differential equation involving a singularity and a radon measure. J. Indian Math. Soc. 86(1–2), 95–117 (2019)
Panda, A., Choudhuri, D., Giri, R.K.: Existence of positive solutions for a singular elliptic problem with critical exponent and measure data. To appear in Rocky Mountain J. Math. https://projecteuclid.org/euclid.rmjm/1608865229
Pekar, S.: Untersuchung uber die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)
Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)
Shieh, T.T., Spector, D.: On a new class of fractional partial differential equations. Adv. Calc. Var. 8(4), 321–336 (2015)
Saoudi, K.: A critical fractional elliptic equation with singular nonlinearities. Fract. Calc. Appl. Anal. 20(6), 1507–1530 (2017)
Saoudi, K., Ghosh, S., Choudhuri, D.: Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity. J. Math. Phys. 60, 101509 (2019)
Sun, Y., Zhang, D.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. 49, 909–922 (2014)
Tang, X., Chen, S.: Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. Adv. Nonlinear Anal. 9(1), 413–437 (2020)
Wang, Y., Yang, Y.: Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator. Bound. Value Probl. 2018, 132 (2018)
Acknowledgements
The author Akasmika Panda thanks the financial assistantship received from the Ministry of Human Resource Development (M.H.R.D.), Govt. of India. Both the authors also acknowledge the facilities received from the Department of mathematics, National Institute of Technology Rourkela. All the authors thank the anonymous referee(s) for their constructive remarks and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Panda, A., Choudhuri, D. & Saoudi, K. A critical fractional choquard problem involving a singular nonlinearity and a radon measure. J. Pseudo-Differ. Oper. Appl. 12, 22 (2021). https://doi.org/10.1007/s11868-021-00382-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11868-021-00382-2