In this paper, we consider the following new Kirchhoff-type equations involving the fractional $p$-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:\begin{eqnarray*} && \left (a+ b\iint _{\mathbb{R}^{2N}} \frac{|u (x)-u (y)|^p}{|x-y|^{N + ps}} {\rm d}x {\rm d}y\right)^{p-1} (-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u\\ &=& \bigg(\int_{\mathbb{R}^N} \frac{|u|^{p^*_{\mu,s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{p^*_{\mu,s}-2}u, \; x \in \mathbb{R}^N, \end{eqnarray*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian with $0 < s < 1 < p$, $0 < \mu < N$, $N > ps$, $a,b>0$, $\lambda>0$ is a parameter, $V:\mathbb{R}^N \to \mathbb{R}^+$ is a potential function, $\theta \in[1, 2^*_{\mu,s})$ and $p^*_{\mu,s}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii's genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the $p$-Laplacian case.
Nemat NYAMORADI
,
Abdolrahman RAZANI
. EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES[J]. Acta mathematica scientia, Series B, 2021
, 41(4)
: 1321
-1332
.
DOI: 10.1007/s10473-021-0418-4
[1] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Functional Analysis, 1973, 14:349-381
[2] Brézis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations//Universitext. New York:Springer, 2011
[3] d'Avenia P, Siciliano G, Squassina M. On fractional Choquard equations. Math Models Methods Appl Sci, 2014, 25(8):1447-1476
[4] Devillanova G, Carlo Marano G. A free fractional viscous oscillator as a forced standard damped vibration. Fractional Calculus and Applied Analysis, 2016, 19(2):319-356
[5] Fiscella A, Pucci P. p-Fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal RWA, 2017, 35:350-378
[6] Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal, 2014, 94:156-170
[7] Guo L, Hu T. Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well. arXiv preprint. 2017, arXiv:1703.08028
[8] Gao F, Shen Z, Yang M. On the critical Choquard equation with potential well. arXiv preprint, 2017, arXiv:1703.01737
[9] Gao F, Yang M. On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation. arXiv:1604.00826v4
[10] Gao F, Yang M. On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents. J Math Anal Appl, 2017, 448(2):1006-1041
[11] Gao F, Yang M. On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. Sci China Math, 2018, 61:1219-1242
[12] Goel D, Sreenadh K. Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. 2019, arXiv:1901.11310v1
[13] Lan F, He X. The Nehari manifold for a fractional critical Choquard equation involving sign-changing weight functions. Nonlinear Anal, 2019, 180:236-263
[14] Li A, Wang P, Wei C. Multiplicity of solutions for a class of Kirchhoff type equations with Hardy-LittlewoodSobolev critical nonlinearity. Appl Math Lett, 2020, 102:106105. DOI:10.1016/j.aml.2019.106105
[15] Lieb E, Loss M. Analysis. 2nd Ed//Grad Stud Math 14. Providence:American Mathematical Society, 2001
[16] Lions P L. The concentration-compactness principle in the calculus of variations. the limit case, part 1. Rev Mat Iberoam, 1985, 1:145-201
[17] Lü D. A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal, 2014, 99:35-48
[18] Ma P, Zhang J. Existence and multiplicity of solutions for fractional Choquard equations. Nonlinear Anal, 2017, 164:100-117
[19] Molica Bisci G, Radulescu V, Servadei R. Variational methods for nonlocal fractional problems. Encyclopedia of Mathematics and its Applications, 162, 2016. Cambridge University Press, ISBN 9781107111943
[20] Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations:Existence, qualitative properties and decay asymptotics. J Functional Anal, 2013, 265(2):153-184
[21] Mukherjee T, Sreenadh K. Fractional Choquard equation with critical nonlinearities. Nonlinear Differ Equat Appl, 2017, 24:63
[22] Mukherjee T, Sreenadh K. On Dirichlet problem for fractional p-Laplacian with singular nonlinearity. Adv Nonlinear Anal, 2016. https://doi.org/10.1515/anona-2016-0100
[23] Mukherjee T, Sreenadh K. Fractional choquard equation with critical nonlinearities. Nonlinear Differential Equations Appl, 2017, 24(6):63, 34 pp
[24] Mukherjee T, Sreenadh K. Positive solutions for nonlinear Choquard equation with singular nonlinearity. Compl Var Ellip Equat, 2017, 62(8):1044-1071
[25] Nyamoradi N, Zaidan L I. Existence and multiplicity of solutions for fractional p-Laplacian SchrödingerKirchhoff type equations. Complex Variables and Elliptic Equations, 2017, 63(2):1-14
[26] Pekar S. Untersuchung über die Elektronentheorie der Kristalle. Berlin:Akademie Verlag, 1954
[27] Pucci P, Xiang M, Zhang B. Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional p-Laplacian. Advances in Calculus of Variations, 2017. DOI:10.1515/acv-2016-0049
[28] Rabinowitz P H. Minimax methods in critical point theory with applications to differential equations//CBMS Reg Conf Series in Math Vol 65. Amer Math Soc Providence, RI, 1986
[29] Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367:67-102
[30] Servadei R, Valdinoci E. A Brezis-Nirenberg result for nonlocal critical equations in low dimension. Commun Pure Appl Anal, 2013, 12:2445-464
[31] Song Y, Shi S. Existence and multiplicity of solutions for Kirchhoff equations with Hardy-LittlewoodSobolev critical nonlinearity. Appl Math Lett, 2019, 92:170-175
[32] Song Y, Shi S. Multiplicity results for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. J Dynamical and Control Systems, 2020, 26:469-480
[33] Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var Partial Differential Equations, 2011, 36:21-41
[34] Wang F, Xiang M. Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent. Elec J Differ Equat, 2016, 306:1-11
[35] Wang F, Xiang M. Multiplicity of solutions for a class of fractional Choquard-Kirchhoff equations involving critical nonlinearity. Anal Math Phys, 2017. https://doi.org/10.1007/s13324-017-0174-8
[36] Wang Y, Yang Y. Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator. Boundary Value Problems, 2018, 132. DOI:10.1186/s13661-018-1050-7
[37] Wang J, Zhang J, Cui Y. Multiple solutions to the Kirchhoff fractional equation involving Hardy-LittlewoodSobolev critical exponent. Boundary Value Problems. 2019, 124. doi:10.1186/s13661-019-1239-4
[38] Willem M. Minimax theorems. Boston:Birkhäuser, 1996
[39] Xiang M Q, Zhang B L, Zhang X. A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in $\mathbb{R}^N$. Adv Nonlinear Stud, 2017, 17(3):611-640
[40] Xiang M, Zhang B, Rǎdulescu V D. Superlinear Schrödinger-Kirchhof type problems involving the fractional p-Laplacian and critical exponent. Adv Nonlinear Anal, 2020, 9:690-709
