We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.
C.O.Alves, F.J.S.A.Corrêa and T.F.Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl.49(1) (2005), 85–93. doi:10.1016/j.camwa.2005.01.008.
2.
A.Ambrosetti and P.H.Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973), 349–381(English). doi:10.1016/0022-1236(73)90051-7.
3.
V.Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Electron. J. Differ. Equ. Journal Profile2016 (2016), Paper No. 151.
4.
V.Ambrosio and T.Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian, Discrete Contin. Dyn. Syst.38 (2018), 5835–5881. doi:10.3934/dcds.2018254.
5.
V.Ambrosio and T.Isernia, On the multiplicity and concentration for p-fractional Schrödinger equations, Appl. Math. Lett.95 (2019), 13–22. doi:10.1016/j.aml.2019.03.010.
6.
V.Ambrosio, T.Isernia and V.D.Radulescu, Concentration of positive solutions for a class of fractional p-Kirchhoff type equations, Proc. R. Soc. Edinb., Sect. A, Math.151 (2021), 601–651. doi:10.1017/prm.2020.32.
7.
D.Applebaum, Lévy processes – From probability to finance and quantum groups, Notices Amer. Math. Soc.51(11) (2004), 1336–1347.
8.
L.Appolloni, G.Molica Bisci and S.Secchi, On critical Kirchhoff problems driven by the fractional Laplacian, Calc. Var. Partial Differential Equations60(6) (2021), Paper No. 209.
9.
L.Brasco and E.Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math.304 (2017), 300–354. doi:10.1016/j.aim.2016.03.039.
10.
L.Brasco, S.Mosconi and M.Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations55(2) (2016), Art. 23.
11.
H.Brézis and E.Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc.88(3) (1983), 486–490. doi:10.1090/S0002-9939-1983-0699419-3.
12.
L.Caffarelli, Non-local diffusions, drifts and games, in: Nonlinear Partial Differential Equations, Abel Symp., Vol. 7, Springer, Heidelberg, 2012, pp. 37–52.
13.
L.Caffarelli and L.Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations32(7–9) (2007), 1245–1260. doi:10.1080/03605300600987306.
14.
L.Caffarelli and E.Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations41(1–2) (2011), 203–240. doi:10.1007/s00526-010-0359-6.
15.
A.Di Castro, T.Kuusi and G.Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire33(5) (2016), 1279–1299. doi:10.1016/j.anihpc.2015.04.003.
16.
E.Di Nezza, G.Palatucci and E.Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.136(5) (2012), 521–573. doi:10.1016/j.bulsci.2011.12.004.
17.
F.Faraci and K.Silva, On the Brezis–Nirenberg problem for a Kirchhoff type equation in high dimension, Calc. Var. Partial Differential Equations60(1) (2021), 22. doi:10.1007/s00526-020-01891-6.
18.
A.Fiscella and E.Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal.94 (2014), 156–170. doi:10.1016/j.na.2013.08.011.
19.
G.Franzina and G.Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N. S.)5(2) (2014), 373–386.
20.
A.Iannizzotto, S.Mosconi and M.Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam.32(4) (2016), 1353–1392. doi:10.4171/RMI/921.
21.
G.Kirchhoff, Vorlesungen über mathematische Physik. Erster Band: Mechanik. Vierte Auflage. Herausgegeben von W. Wien. Mit 18 Figuren im Text, Leipzig: B. G. Teubner. X und 464 S. gr. , 1897.
22.
E.H.Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2)118(2) (1983), 349–374. doi:10.2307/2007032.
23.
E.Lindgren and P.Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations49(1–2) (2014), 795–826. doi:10.1007/s00526-013-0600-1.
24.
P.-L.Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana1(1) (1985), 145–201. doi:10.4171/RMI/6.
25.
P.-L.Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana1(2) (1985), 45–121. doi:10.4171/RMI/12.
26.
R.Metzler and J.Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A37(31) (2004), R161–R208. doi:10.1088/0305-4470/37/31/R01.
27.
X.Mingqi, G.Molica Bisci, G.Tian and B.Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity29(2) (2016), 357–374. doi:10.1088/0951-7715/29/2/357.
28.
S.Mosconi, K.Perera, M.Squassina and Y.Yang, The Brezis–Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations55(4) (2016), Art. 105.
29.
S.Mosconi and M.Squassina, Nonlocal problems at nearly critical growth, Nonlinear Anal.136 (2016), 84–101. doi:10.1016/j.na.2016.02.012.
30.
K.Perera, M.Squassina and Y.Yang, Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr.289(2–3) (2016), 332–342. doi:10.1002/mana.201400259.
31.
P.Pucci, M.Xiang and B.Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in , Calc. Var. Partial Differential Equations54(3) (2015), 2785–2806. doi:10.1007/s00526-015-0883-5.
32.
H.Royden and P.M.Fitzpatrick, Real Analysis, Prentice Hall, New York, NY, 2010(English).
33.
A.S.Vasudeva Murthy, On the string equation of Narasimha, in: Connected at Infinity. II, Texts Read. Math., Vol. 67, Hindustan Book Agency, New Delhi, 2013, pp. 58–84.
34.
M.Xiang, B.Zhang and M.Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl.424(2) (2015), 1021–1041. doi:10.1016/j.jmaa.2014.11.055.
35.
M.Xiang, B.Zhang and V.D.Rădulescu, Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal.9(1) (2020), 690–709. doi:10.1515/anona-2020-0021.
