과제정보
The authors would like to thank the anonymous referees for their carefully reading this paper and their useful comments.
참고문헌
- D. Applebaum, Levy processes and stochastic calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511809781
- D. Averna, S. Tersian, and E. Tornatore, On the existence and multiplicity of solutions for Dirichlet's problem for fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 253-266. https://doi.org/10.1515/fca-2016-0014
- L. Brasco, E. Lindgren, and E. Parini, The fractional Cheeger problem, Interfaces Free Bound. 16 (2014), no. 3, 419-458. https://doi.org/10.4171/IFB/325
- L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323-355. https://doi.org/10.1515/acv-2015-0007
- H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490. https://doi.org/10.2307/2044999
-
H. Brezis and L. Nirenberg,
$H^1\;versus\;C^1$ local minimizers, C. R. Acad. Sci. Paris Ser. I Math. 317 (1993), no. 5, 465-472. - F. Brock, L. Iturriaga, and P. Ubilla, A multiplicity result for the p-Laplacian involving a parameter, Ann. Henri Poincare 9 (2008), no. 7, 1371-1386. https://doi.org/10.1007/s00023-008-0386-4
- D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004), no. 2, 521-537. https://doi.org/10.1016/j.jde.2004.03.005
- W. Chen, S. Mosconi, and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018), no. 11, 3065-3114. https://doi.org/10.1016/j.jfa.2018.02.020
- A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), no. 1, 225-236. https://doi.org/10.1016/j.jmaa.2004.03.034
- D. G. de Figueiredo, J.-P. Gossez, and P. Ubilla, Local \superlinearity" and \sublinearity" for the p-Laplacian, J. Funct. Anal. 257 (2009), no. 3, 721-752. https://doi.org/10.1016/j.jfa.2009.04.001
- J. Garcia Azorero and I. Peral Alonso, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 (1994), no. 3, 941-957. https://doi.org/10.1512/iumj.1994.43.43041
- J. P. Garcia Azorero, I. Peral Alonso, and J. J. Manfredi, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), no. 3, 385-404. https://doi.org/10.1142/S0219199700000190
- A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fract. Differ. Calc. 6 (2016), no. 2, 201-217. https://doi.org/10.7153/fdc-06-13
- A. Ghanmi and K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator, Complex Var. Elliptic Equ. 61 (2016), no. 9, 1199-1216. https://doi.org/10.1080/17476933.2016.1154548
- N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), no. 5, 321-330. https://doi.org/10.1016/S0294-1449(16)30313-4
- N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5703-5743. https://doi.org/10.1090/S0002-9947-00-02560-5
-
J. Giacomoni and K. Saoudi,
$W^-1}_-0}^-,p}\;versus\;C^1$ local minimizers for a singular and critical functional, J. Math. Anal. Appl. 363 (2010), no. 2, 697-710. https://doi.org/10.1016/j.jmaa.2009.10.012 - T.-S. Hsu, Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Nonlinear Anal. 74 (2011), no. 12, 3934-3944. https://doi.org/10.1016/j.na.2011.02.036
- A. Iannizzotto, S. Liu, K. Perera, and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), no. 2, 101-125. https://doi.org/10.1515/acv-2014-0024
- A. Iannizzotto, S. Mosconi, and M. Squassina, Global Holder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), no. 4, 1353-1392. https://doi.org/10.4171/RMI/921
- R.-T. Jiang and C.-L. Tang, Semilinear elliptic problems involving Hardy-Sobolev-Maz'ya potential and Hardy-Sobolev critical exponents, Electron. J. Differential Equations 2016 (2016), Paper No. 12, 8 pp.
- S. Liang and J. Zhang, Multiplicity of solutions for a class of quasi-linear elliptic equation involving the critical Sobolev and Hardy exponents, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 1, 55-67. https://doi.org/10.1007/s00030-009-0039-4
- S. Mosconi, K. Perera, M. Squassina, and Yang Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 105, 25 pp. https://doi.org/10.1007/s00526-016-1035-2
- E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004
- K. Perera, M. Squassina, and Y. Yang, Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr. 289 (2016), no. 2-3, 332-342. https://doi.org/10.1002/mana.201400259
- K. Perera and W. Zou, p-Laplacian problems involving critical Hardy-Sobolev exponents, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 3, Art. 25, 16 pp. https://doi.org/10.1007/s00030-018-0517-7
-
K. Saoudi, On
$W^-s,p}\;vs.\;C^1$ local minimizers for a critical functional related to fractional p-Laplacian, Appl. Anal. 96 (2017), no. 9, 1586-1595. https://doi.org/10.1080/00036811.2017.1307964 - K. Saoudi, A critical fractional elliptic equation with singular nonlinearities, Fract. Calc. Appl. Anal. 20 (2017), no. 6, 1507-1530. https://doi.org/10.1515/fca-2017-0079
- K. Saoudi and M. Kratou, Existence of multiple solutions for a singular and quasilinear equation, Complex Var. Elliptic Equ. 60 (2015), no. 7, 893-925. https://doi.org/10.1080/17476933.2014.981169
- R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887-898. https://doi.org/10.1016/j.jmaa.2011.12.032
- R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105-2137. https://doi.org/10.3934/dcds.2013.33.2105
- G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), no. 3, 281-304. https://doi.org/10.1016/S0294-1449(16)30238-4
- C.Wang and Y.-Y. Shang, Existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 451 (2017), no. 2, 1198-1215. https://doi.org/10.1016/j.jmaa.2017.02.063
- L. Wang, Q. Wei, and D. Kang, Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardy-type term, Nonlinear Anal. 74 (2011), no. 2, 626-638. https://doi.org/10.1016/j.na.2010.09.017
- Y. Yang, The brezis Nirenberg problem for the fractional p-laplacian involving critical hardy sobolev exponents, https://arxiv.org/abs/1710.04654.