In this paper, we study a fractional Schrödinger–Poisson system with p-Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.
L.Appolloni and S.Secchi, Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differ. Equ.286 (2021), 248–283. doi:10.1016/j.jde.2021.03.016.
2.
M.B.Benboubker, H.Benkhalou, H.Hjiaj and I.Nyanquini, Entropy solutions for elliptic Schrödinger type equations under Fourier boundary conditions, Rend. Circ. Mat. Palermo (2)72(4) (2023), 2831–2855. doi:10.1007/s12215-022-00822-y.
3.
V.Benci and D.Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal.11(2) (1998), 283–293. doi:10.12775/TMNA.1998.019.
4.
V.Benci and D.Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations, Rev. Math. Phys.14(4) (2002), 409–420. doi:10.1142/S0129055X02001168.
5.
J.Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Vol. 121, Cambridge University Press, Cambridge, 1996.
S.T.Chen, V.D.Rădulescu and X.H.Tang, Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth, Math. Z.306(3) (2024), 50.
9.
S.T.Chen, M.H.Shu, X.H.Tang and L.X.Wen, Planar Schrödinger–Poisson system with critical exponential growth in the zero mass case, J. Differ. Equ.327 (2022), 448–480. doi:10.1016/j.jde.2022.04.022.
10.
T.D’Aprile and D.Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud.4(3) (2004), 307–322. doi:10.1515/ans-2004-0305.
11.
Y.Du, J.B.Su and C.Wang, The Schrödinger–Poisson system with p-Laplacian, Appl. Math. Lett.120 (2021), 1–7. doi:10.1016/j.aml.2021.107342.
12.
X.J.Feng, Nontrivial solution for Schrödinger–Poisson equations involving a fractional nonlocal operator via perturbation methods, Z. Angew. Math. Phys.67(74) (2016), 1–10.
13.
G.Gilboa and S.Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul.7(3) (2008), 1005–1028. doi:10.1137/070698592.
14.
K.P.Ho, Two-weight norm inequalities for rough fractional integral operators on Morrey spaces, Opuscula Math.44(1) (2024), 67–77. doi:10.7494/OpMath.2024.44.1.67.
15.
L.Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal.28(10) (1997), 1633–1659. doi:10.1016/S0362-546X(96)00021-1.
16.
L.Jeanjean and S.Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differ. Equ.11(7) (2006), 813–840.
17.
N.Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A268(4–6) (2000), 298–305. doi:10.1016/S0375-9601(00)00201-2.
18.
C.Lei, J.Lei and H.Suo, Groundstate for the Schrödinger–Poisson–Slater equation involving the Coulomb–Sobolev critical exponent, Adv. Nonlinear Anal.12(1) (2023), 20220299, 17 pp.
19.
K.X.Li, Existence of non-trivial solutions for nonlinear fractional Schrödinger–Poisson equations, Appl. Math. Lett.72 (2017), 1–9. doi:10.1016/j.aml.2017.03.023.
20.
H.J.Luo and Z.T.Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differ. Equ.59(4) (2020), 1–35. doi:10.1007/s00526-020-01814-5.
21.
R.Metzler and J.Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep.339(1) (2000), 1–77. doi:10.1016/S0370-1573(00)00070-3.
22.
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), 161–208. doi:10.1088/0305-4470/37/31/R01.
23.
E.D.Nezza, G.Patalluci 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.
24.
P.H.Rabinowitz, Minmax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math, Vol. 65, Amer. Math. Soc, Providence, RI, 1986.
25.
E.M.Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
26.
K.M.Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent, J. Differ. Equ.261(6) (2016), 3061–3106. doi:10.1016/j.jde.2016.05.022.
27.
L.Torres and E.César, Existence and symmetry result for fractional p-Laplacian in , Commun. Pure Appl. Anal.16(1) (2017), 99–113.
28.
L.Wang, V.D.Rădulescu and B.Zhang, Infinitely many solutions for fractional Kirchhoff–Schrödinger–Poisson systems, J. Math. Phys.60(1) (2019), 011506, 18 pp.
29.
X.Wang, F.Chen and F.Liao, Existence and nonexistence of nontrivial solutions for the Schrödinger–Poisson system with zero mass potential, Adv. Nonlinear Anal.12(1) (2023), 20220319, 12 pp.
M.Willem, Functional Analysis. Fundamentals and Applications, Birkhäuser/Springer, New York, 2013.
32.
Y.Y.Yu, F.K.Zhao and L.G.Zhao, Positive and signchanging least energy solutions for a fractional Schrödinger–Poisson system with critical exponent, Appl. Anal.99(13) (2020), 2229–2257. doi:10.1080/00036811.2018.1557325.
33.
J.J.Zhang, J.M.Do Ó and M.Squassina, Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud.16(1) (2016), 15–30. doi:10.1515/ans-2015-5024.
34.
M.Zhao, Y.Q.Song and D.D.Repovš, On the p-fractional Schrödinger–Kirchhoff equations with electromagnetic fields and the Hardy–Littlewood–Sobolev nonlinearity, Demonstr. Math.57(1) (2024), dema–2023–0124.
35.
J.B.Zuo, C.G.Liu and C.Vetro, Normalized solutions to the fractional Schrödinger equation with potential, Mediterr. J. Math.20(4) (2023), 12.