Existence and multiplicity of solutions for critical Choquard-Kirchhoff type equations with variable growth

Lulu Tao; Rui He; Sihua Liang; Rui Niu; Lulu Tao; Rui He; Sihua Liang; Rui Niu

doi:10.3934/math.2023156

AIMS Mathematics

2023, Volume 8, Issue 2: 3026-3048. doi: 10.3934/math.2023156

Previous Article Next Article

Research article

Existence and multiplicity of solutions for critical Choquard-Kirchhoff type equations with variable growth

1.
College of Mathematics, Changchun Normal University, Changchun 130032, China
2.
College of Science, Heilongjiang Institute of Technology, Harbin 150050, China

Received: 15 September 2022 Revised: 26 October 2022 Accepted: 04 November 2022 Published: 15 November 2022
MSC : 35J20, 35J60, 35J62

We prove the existence and multiplicity of solutions for a class of Choquard-Kirchhoff type equations with variable exponents and critical reaction. Because the appearance of the critical reaction, we deal with the lack of compactness by using the concentration-compactness principle. In particular, we discuss the main results in non-degenerate and degenerate cases. And we apply combination of Krasnoselskii genus and the Hardy-Littlewood-Sobolev inequality to get the results of existence and multiplicity.
- Choquard equation,
- critical nonlinearity,
- concentration-compactness principles,
- variational methods
Citation: Lulu Tao, Rui He, Sihua Liang, Rui Niu. Existence and multiplicity of solutions for critical Choquard-Kirchhoff type equations with variable growth[J]. AIMS Mathematics, 2023, 8(2): 3026-3048. doi: 10.3934/math.2023156

Related Papers:

Abstract

We prove the existence and multiplicity of solutions for a class of Choquard-Kirchhoff type equations with variable exponents and critical reaction. Because the appearance of the critical reaction, we deal with the lack of compactness by using the concentration-compactness principle. In particular, we discuss the main results in non-degenerate and degenerate cases. And we apply combination of Krasnoselskii genus and the Hardy-Littlewood-Sobolev inequality to get the results of existence and multiplicity.

References

[1]	C. O. Alves, J. L. P. Barreiro, Existence and multiplicity of solutions for a $p(x)$-Laplacian equation with critical growth, J. Math. Anal. Appl., 403 (2013), 143–154. https://doi.org/10.1016/j.jmaa.2013.02.025 doi: 10.1016/j.jmaa.2013.02.025
[2]	C. O. Alves, L. S. Tavares, A Hardy-Littlewood-Sobolev-type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent, Mediterr. J. Math., 16 (2019), 1–27. https://doi.org/10.1007/S00009-019-1316-Z doi: 10.1007/S00009-019-1316-Z
[3]	A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
[4]	G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699–714. http://doi.org/10.1016/j.na.2015.06.014 doi: 10.1016/j.na.2015.06.014
[5]	A. K. Ben-Naouma, C. Troestler, M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal., 26 (1996), 823–833. https://doi.org/10.1016/0362-546X(94)00324-B doi: 10.1016/0362-546X(94)00324-B
[6]	G. Bianchi, J. Chabrowski, A. Szulkin, Symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal., 25 (1995), 41–59. https://doi.org/10.1016/0362-546X(94)E0070-W doi: 10.1016/0362-546X(94)E0070-W
[7]	J. F. Bonder, A. Silva, Concentration-compactness principle for variable exponent spaces and applications, Electron. J. Differ. Equ., 141 (2010), 1–18. https://doi.org/10.48550/arXiv.0906.1922 doi: 10.48550/arXiv.0906.1922
[8]	H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
[9]	M. Caponi, P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099–2129. https://doi.org/10.1007/s10231-016-0555-x doi: 10.1007/s10231-016-0555-x
[10]	J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Equ., 3 (1995), 493–512. https://doi.org/10.1007/BF01187898 doi: 10.1007/BF01187898
[11]	Y. Chen, S. Levine, R. Rao, Functionals with $p(x)$ growth in image processing, SIAM J. Appl. Math., 66 (2006), 1383–1406. https://doi.org/10.1137/050624522 doi: 10.1137/050624522
[12]	P. D'Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247–262. https://doi.org/10.1007/BF02100605 doi: 10.1007/BF02100605
[13]	L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer-Verlag, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
[14]	X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617 doi: 10.1006/jmaa.2000.7617
[15]	Y. Fu, The principle of concentration compactness in $L^{p(x)}$ spaces and its application, Nonlinear Anal., 71 (2009), 1876–1892. https://doi.org/10.1016/j.na.2009.01.023 doi: 10.1016/j.na.2009.01.023
[16]	Y. Fu, X. Zhang, Multiple solutions for a class of $p(x)$-Laplacian equations in $\mathbb{R}^N$ involving the critical exponent, Proc. Roy. Soc., 466 (2010), 1667–1686. https://doi.org/10.1098/rspa.2009.0463 doi: 10.1098/rspa.2009.0463
[17]	H. Fröhlich, Theory of electrical breakdown in ionic crystals, Proc. Roy. Soc. Edinburgh Sect., 160 (1937), 230–241. https://doi.org/10.1098/rspa.1937.0106 doi: 10.1098/rspa.1937.0106
[18]	K. Ho, Y. H. Kim, A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian, Nonlinear Anal., 188 (2019), 179–201. https://doi.org/10.1016/j.na.2019.06.001 doi: 10.1016/j.na.2019.06.001
[19]	K. Ho, Y. H. Kim, I. Sim, Existence results for Schr$\ddot{ o }$dinger $p(\cdot)$-Laplace equations involving critical growth in ${\mathbb {R}}^N$, arXiv, 2018. https://doi.org/10.48550/arXiv.1807.03961
[20]	S. Liang, J. Zhang, Multiple solutions for a noncooperative $p(x)$-Laplacian equations in $\mathbb{R}^N$ involving the critical exponent, J. Math. Anal. Appl., 403 (2013), 344–356. https://doi.org/10.1016/j.jmaa.2013.01.003 doi: 10.1016/j.jmaa.2013.01.003
[21]	S. Liang, J. Zhang, Infinitely many small solutions for the $p(x)$-Laplacian operator with nonlinear boundary conditions, Ann. Mat. Pura Appl., 192 (2013), 1–16. https://doi.org/10.1007/s10231-011-0209-y doi: 10.1007/s10231-011-0209-y
[22]	S. Liang, G. Molica Bisci, B. Zhang, Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponents, Nonlinear Anal. Model. Control, 27 (2022), 1–20. https://doi.org/10.15388/namc.2022.27.26575 doi: 10.15388/namc.2022.27.26575
[23]	S. Liang, D. Repov$\breve{s}$, B. Zhang, Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities, Math. Meth. Appl. Sci., 43 (2020), 2473–2490. https://doi.org/10.1002/mma.6057 doi: 10.1002/mma.6057
[24]	S. Liang, P. Pucci, B. Zhang, Existence and multiplicity of solutions for critical nonlocal equations with variable exponents, Appl. Anal., 2022, 1–24. https://doi.org/10.1080/00036811.2022.2107916 doi: 10.1080/00036811.2022.2107916
[25]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977) 93–105. https://doi.org/10.1002/sapm197757293 doi: 10.1002/sapm197757293
[26]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, part I, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109–145. https://doi.org/10.1016/S0294-1449(16)30428-0 doi: 10.1016/S0294-1449(16)30428-0
[27]	P. L. Lions, The concentration compactness principle in the calculus of variations. The limit case, part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1985), 45–121. https://doi.org/10.4171/RMI/12 doi: 10.4171/RMI/12
[28]	B. B. V. Maia, On a class of $p(x)$-Choquard equations with sign-changing potential and upper critical growth, Rend. Circ. Mat. Palermo, II. Ser., 70 (2021), 1175–1199. https://doi.org/10.1007/s12215-020-00553-y doi: 10.1007/s12215-020-00553-y
[29]	J. Mawhin, G. Molica Bisci, A Brezis-Nirenberg type result for a nonlocal fractional operator, J. Lond. Math. Soc., 95 (2017), 73–93. https://doi.org/10.1112/jlms.12009 doi: 10.1112/jlms.12009
[30]	X. Mingqi, G. Molica Bisci, G. Tian, B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357–374. https://doi.org/10.1088/0951-7715/29/2/357 doi: 10.1088/0951-7715/29/2/357
[31]	I. M. Moroz, R. Penrose, P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quant. Grav., 15 (1998), 2733. https://doi.org/10.1088/0264-9381/15/9/019 doi: 10.1088/0264-9381/15/9/019
[32]	V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005. https://doi.org/10.1142/S0219199715500054 doi: 10.1142/S0219199715500054
[33]	V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153–184. https://doi.org/10.1016/j.jfa.2013.04.007 doi: 10.1016/j.jfa.2013.04.007
[34]	W. Orlicz, Über konjugierte exponentenfolgen, Studia Math., 3 (1931), 200–212. https://doi.org/10.4064/sm-3-1-200-211 doi: 10.4064/sm-3-1-200-211
[35]	S. Pekar, Untersuchung über die elektronentheorie der Kristalle, Akad. Verlag, 1954. https://doi.org/10.1515/9783112649305
[36]	R. Penrose, On Gravity's role in quantum state reduction, Gen. Relat. Gravit., 28 (1996), 581–600. https://doi.org/10.1007/BF02105068 doi: 10.1007/BF02105068
[37]	P. Pucci, M. Xiang, B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27–55. https://doi.org/10.1515/anona-2015-0102 doi: 10.1515/anona-2015-0102
[38]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, 1986. https://doi.org/10.1090/cbms/065
[39]	V. R$\breve{ a }$dulescu, D. Repov$\breve{ s }$, Partial differential equations with variable exponents, 1 Ed., CRC Press, Boca Raton, 2015. https://doi.org/10.5772/15954
[40]	M. Ruzicka, Electrorheological fluids modeling and mathematical theory, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/bfb0104029
[41]	X. Shi, Y. Zhao, H. Chen, Existence of solutions for nonhomogeneous Choquard equations involving $p$-Laplacian, Mathematics, 7 (2019), 871. https://doi.org/10.3390/math7090871 doi: 10.3390/math7090871
[42]	Y. Song, S. Shi, On a degenerate $p$-fractional Kirchhoff equations involving critical Sobolev-Hardy nonlinearities, Mediterr. J. Math., 15 (2018), 1–18. https://doi.org/10.1007/s00009-017-1062-z doi: 10.1007/s00009-017-1062-z
[43]	X. Sun, Y. Song, S. Liang, On the critical Choquard-Kirchhoff problem on the Heisenberg group, Adv. Nonlinear Anal., 12 (2023), 210–236. https://doi.org/10.1515/anona-2022-0270 doi: 10.1515/anona-2022-0270
[44]	L. Wang, K. Xie, B. Zhang, Existence and multiplicity of solutions for critical Kirchhoff-type $p$-Laplacian problems, J. Math. Anal. Appl., 458 (2018), 361–378. https://doi.org/10.1016/j.jmaa.2017.09.008 doi: 10.1016/j.jmaa.2017.09.008
[45]	F. Gao, M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219–1242. https://doi.org/10.1007/s11425-016-9067-5 doi: 10.1007/s11425-016-9067-5
[46]	F. Gao, M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006–1041. https://doi.org/10.1016/j.jmaa.2016.11.015 doi: 10.1016/j.jmaa.2016.11.015
[47]	Y. Zhang, X. Tang, V. R$\breve{ {\rm{a}} }$dulescu, High perturbations of Choquard equations with critical reaction and variable growth, Proc. Amer. Math. Soc., 149 (2021), 3819–3835. http://dx.doi.org/10.1090/proc/15469 doi: 10.1090/proc/15469

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)