Abstract
In this paper, using variational methods, we study the existence of ground states solutions to the modified fractional Schrödinger equations with a generalized Choquard nonlinearity.
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REFERENCES
S. Pekar, Untersuchung über die Elektronentheorie der Kristalle (Akademie Verlag, Berlin, 1954).
P. d’Avenia, G. Siciliano, and M. Squassina, ‘‘On fractional Choquard equations,’’ Math. Models Methods Appl. Sci. 25, 1447–1476 (2014). https://doi.org/10.1142/S0218202515500384
L. Guo and T. Hu, ‘‘Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well’’ (2017). arXiv:1703.08028 [math.AP]
F. Gao, Z. Shen, and M. Yang, ‘‘On the critical Choquard equation with potential well’’ (2017). arXiv:1703.01737 [math.AP]
T. Mukherjee and K. Sreenadh, ‘‘Fractional Choquard equation with critical nonlinearities,’’ Nonlinear Differ. Equations Appl. 24, 63 (2017). https://doi.org/10.1007/s00030-017-0487-1
T. Mukherjee and K. Sreenadh, ‘‘On Dirichlet problem for fractional \(p\)-Laplacian with singular nonlinearity,’’ Adv. Nonlinear Anal. 8, 52–72 (2019). https://doi.org/10.1515/anona-2016-0100.
F. Lan and X. He, ‘‘The Nehari manifold for a fractional critical Choquard equation involving sign-changing weight functions,’’ Nonlinear Anal. 180, 236–263 (2019). https://doi.org/10.1016/j.na.2018.10.010
P. Ma and J. Zhang, ‘‘Existence and multiplicity of solutions for fractional Choquard equations,’’ Nonlinear Anal. 164, 100–117 (2017). https://doi.org/10.1016/j.na.2017.07.011
P. Pucci, M. Xiang, and B. Zhang, ‘‘Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional \(p\)-Laplacian,’’ Adv. Calculus Var. 12, 253–275 (2017). https://doi.org/10.1515/acv-2016-0049
F. Wang and M. Xiang, ‘‘Multiplicity of solutions for a class of fractional Choquard–Kirchhoff equations involving critical nonlinearity,’’ Anal. Math. Phys. 9, 1–16 (2017). https://doi.org/10.1007/s13324-017-0174-8
J. Wang, J. Zhang, and Y. Cui, ‘‘Multiple solutions to the Kirchhoff fractional equation involving Hardy–Littlewood–Sobolev critical exponent,’’ Boundary Value Problems 2019, 124 (2019). https://doi.org/10.1186/s13661-019-1239-4
Y. Wang and Y. Yang, ‘‘Bifurcation results for the critical Choquard problem involving fractional \(p\)-Laplacian operator,’’ Boundary Value Problems 2018, 132 (2018). https://doi.org/10.1186/s13661-018-1050-7
T. Mukherjee and K. Sreenadh, ‘‘Fractional Choquard equation with critical nonlinearities,’’ Nonlinear Differ. Equations Appl. 24, 63 (2017). https://doi.org/10.1007/s00030-017-0487-1
F. Gao and M. Yang, ‘‘On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation,’’ Sci. China Math. 61, 1219–1242 (2018). https://doi.org/10.1007/s11425-016-9067-5
R. Servadei and E. Valdinoci, ‘‘The Brezis–Nirenberg result for the fractional Laplacian,’’ Trans. Amer. Math. Soc. 367, 67–102 (2015). https://doi.org/10.1090/S0002-9947-2014-05884-4
R. Servadei and E. Valdinoci, ‘‘A Brezis–Nirenberg result for nonlocal critical equations in low dimension,’’ Commun. Pure Appl. Anal. 12, 2445–2464 (2013). https://doi.org/10.3934/cpaa.2013.12.2445
J. Tan, ‘‘The Brezis–Nirenberg type problem involving the square root of the Laplacian,’’ Calculus Var. Partial Differ. Equations 36, 21–41 (2011). https://doi.org/10.1007/s00526-010-0378-3
L. Shao and Y. Wang, ‘‘Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity,’’ Open Math. 19, 259–267 (2021). https://doi.org/10.1515/math-2021-0025
J. Zhang and C. Ji, ‘‘Ground state solutions for a generalized quasilinear Choquard equation,’’ Math. Meth. Appl Sci. 44, 6048–6055 (2021). https://doi.org/10.1002/mma.7169
Y. Song and S. Shi, ‘‘Existence and multiplicity of solutions for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity,’’ Appl. Math. Lett. 92, 170–175 (2019). https://doi.org/10.1016/j.aml.2019.01.017
G. Devillanova and G. Carlo Marano, ‘‘A free fractional viscous oscillator as a forced standard damped vibration,’’ Fract. Calculus Appl. Anal. 19, 319–356 (2016). https://doi.org/10.1515/fca-2016-0018
A. Fiscella and E. Valdinoci, ‘‘A critical Kirchhoff type problem involving a nonlocal operator,’’ Nonlinear Anal. 94, 156–170 (2014). https://doi.org/10.1016/j.na.2013.08.011
F. Gao and M. Yang, ‘‘On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents,’’ J. Math. Anal. Appl. 448, 1006–1041 (2017). https://doi.org/10.1016/j.jmaa.2016.11.015
D. Goel and K. Sreenadh, ‘‘Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity,’’ Nonlinear Anal. 186, 162–186 (2019). https://doi.org/10.1016/j.na.2019.01.035
T. Mukherjee and K. Sreenadh, ‘‘Positive solutions for nonlinear Choquard equation with singular nonlinearity,’’ Complex Var. Elliptic Equations 62, 1044–1071 (2017). https://doi.org/10.1080/17476933.2016.1260559
A. Li, P. Wang, and C. Wei, ‘‘Multiplicity of solutions for a class of Kirchhoff type equations with Hardy–Littlewood–Sobolev critical nonlinearity,’’ Appl. Math. Lett. 102, 106105 (2020). https://doi.org/10.1016/j.aml.2019.106105
G. Molica Bisci, V. Radulescu, and R. Servadei, ‘‘Variational methods for nonlocal fractional problems,’’ in Encyclopedia of Mathematics and Its Applications, vol. 162 (Cambridge Univ. Press, Cambridge, 2016).
V. Moroz and J. Van Schaftingen, ‘‘Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics,’’ J. Funct. Anal. 265 (2), 153–184 (2013). https://doi.org/10.1016/j.jfa.2013.04.007
F. Wang and M. Xiang, ‘‘Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent,’’ Electron. J. Differ. Equations 306, 1–11 (2016).
X. Yang, X. Tang, and G. Gu, ‘‘Concentration behavior of ground states for a generalized quasilinear Choquard equation,’’ Math. Methods Appl. Sci. 43, 3569–3585 (2020). https://doi.org/10.1002/mma.6138
F. Gao and J. Zhou, ‘‘Semiclassical states for critical Choquard equations with critical frequency,’’ Topol. Methods Nonlinear Anal. 57, 107–133 (2021). https://doi.org/10.12775/TMNA.2020.001
X. Wu, W. Zhang, and X. Zhou, ‘‘Ground state solutions for a modified fractional Schrödinger equation with critical exponent,’’ Math. Methods Appl. Sci. , (2020). https://doi.org/10.1002/mma.6090
N. Nyamoradi and L. I. Zaidan, ‘‘Existence and multiplicity of solutions for fractional \(p\)-Laplacian Schrödinger–Kirchhoff type equations,’’ Complex Var. Elliptic Equations 63, 346–359 (2017). https://doi.org/10.1080/17476933.2017.1310851
P. Pucci, M. Xiang, and B. Zhang, ‘‘Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in \(\mathbb{R}^{N}\),’’ Calculus Var. 54, 2785–2806 (2015). https://doi.org/10.1007/s00526-015-0883-5
W. Chen, S. Mosconi, and M. Squassina, ‘‘Nonlocal problems with critical Hardy nonlinearity,’’ J. Funct. Anal. 275, 3065–3114 (2018). https://doi.org/10.1016/j.jfa.2018.02.020
E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14 (American Mathematical Society, Providence, 2001).
W. Chen, ‘‘Critical fractional p-Kirchhoff type problem with a generalized Choquard nonlinearity,’’ J. Math Phys. 59, 121502 (2018). https://doi.org/10.1063/1.5052669
H. Brézis and E. Lieb, ‘‘A relation between pointwise convergence of functions and convergence of functionals,’’ Proc. Am. Math. Soc. 88, 486–490 (1983). https://doi.org/10.1090/S0002-9939-1983-0699419-3
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Dehsari, I., Nyamoradi, N. Ground States Solutions for a Modified Fractional Schrödinger Equation with a Generalized Choquard Nonlinearity. J. Contemp. Mathemat. Anal. 57, 131–144 (2022). https://doi.org/10.3103/S1068362322030025
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DOI: https://doi.org/10.3103/S1068362322030025