Existence and multiplicity of standing wave solutions for perturbed fractional <i>p</i>-Laplacian systems involving critical exponents

Shulin Zhang; Shulin Zhang

doi:10.3934/math.2023048

AIMS Mathematics

2023, Volume 8, Issue 1: 997-1013. doi: 10.3934/math.2023048

Previous Article Next Article

Research article

Existence and multiplicity of standing wave solutions for perturbed fractional p-Laplacian systems involving critical exponents

Shulin Zhang ^{1,2
,
,}

1.
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
2.
School of Mathematics, Xuzhou Vocational Technology Academy of Finance and Economics, Xuzhou 221116, China

Received: 31 August 2022 Revised: 29 September 2022 Accepted: 07 October 2022 Published: 14 October 2022
MSC : 35B33, 35J50, 35J60

In this paper, we investigate the existence of standing wave solutions to the following perturbed fractional p-Laplacian systems with critical nonlinearity

$ \begin{equation*} \left\{ \begin{aligned} &\varepsilon^{ps}(-\Delta)^{s}_{p}u + V(x)|u|^{p-2}u = K(x)|u|^{p^{*}_{s}-2}u + F_{u}(x, u, v), \; x\in \mathbb{R}^{N}, \\ &\varepsilon^{ps}(-\Delta)^{s}_{p}v + V(x)|v|^{p-2}v = K(x)|v|^{p^{*}_{s}-2}v + F_{v}(x, u, v), \; x\in \mathbb{R}^{N}. \end{aligned} \right. \end{equation*} $

Under some proper conditions, we obtain the existence of standing wave solutions $ (u_{\varepsilon}, v_{\varepsilon}) $ which tend to the trivial solutions as $ \varepsilon\rightarrow 0 $. Moreover, we get $ m $ pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results.
- perturbed fractional p-Laplacian systems,
- critical growth,
- variational methods,
- standing wave solutions
Citation: Shulin Zhang. Existence and multiplicity of standing wave solutions for perturbed fractional p-Laplacian systems involving critical exponents[J]. AIMS Mathematics, 2023, 8(1): 997-1013. doi: 10.3934/math.2023048

Related Papers:

Abstract

In this paper, we investigate the existence of standing wave solutions to the following perturbed fractional p-Laplacian systems with critical nonlinearity

$ \begin{equation*} \left\{ \begin{aligned} &\varepsilon^{ps}(-\Delta)^{s}_{p}u + V(x)|u|^{p-2}u = K(x)|u|^{p^{*}_{s}-2}u + F_{u}(x, u, v), \; x\in \mathbb{R}^{N}, \\ &\varepsilon^{ps}(-\Delta)^{s}_{p}v + V(x)|v|^{p-2}v = K(x)|v|^{p^{*}_{s}-2}v + F_{v}(x, u, v), \; x\in \mathbb{R}^{N}. \end{aligned} \right. \end{equation*} $

Under some proper conditions, we obtain the existence of standing wave solutions $ (u_{\varepsilon}, v_{\varepsilon}) $ which tend to the trivial solutions as $ \varepsilon\rightarrow 0 $. Moreover, we get $ m $ pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results.

References

[1]	D. Applebaum, Lévy processes-from probability theory to finance and quantum groups, Notices of the American Mathematical Society, 51 (2004), 1336–1347.
[2]	H. Alsulami, M. Kirane, S. Alhodily, T. Saeed, N. Nyamoradi, Existence and multiplicity of sulutions to fractional p-Laplacian system with concave-convex nonlinearities, Bull. Math. Sci., 10 (2020), 2050007. https://doi.org/10.1142/S1664360720500071 doi: 10.1142/S1664360720500071
[3]	V. Benci, On critical point theory of indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533–572. https://doi.org/10.1090/S0002-9947-1982-0675067-X doi: 10.1090/S0002-9947-1982-0675067-X
[4]	H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490. https://doi.org/10.1090/S0002-9939-1983-0699419-3 doi: 10.1090/S0002-9939-1983-0699419-3
[5]	J. Byeon, Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems, J. Differ. Equations, 217 (2005), 257–281. https://doi.org/10.1016/j.jde.2005.07.008 doi: 10.1016/j.jde.2005.07.008
[6]	L. Caffarelli, Nonlocal equations, drifts and games, In: Nonlinear partial differential equations, Berlin, Heidelberg: Springer, 2012, 37–52. https://doi.org/10.1007/978-3-642-25361-4_3
[7]	G. Chen, Y. Zhang, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359–2376. https://doi.org/10.3934/cpaa.2014.13.2359 doi: 10.3934/cpaa.2014.13.2359
[8]	W. Chen, S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Anal. Real, 27 (2016), 80–92. https://doi.org/10.1016/j.nonrwa.2015.07.009 doi: 10.1016/j.nonrwa.2015.07.009
[9]	W. Chen, M. Squassina, Critical nonlocal systems with concave-convex powers, Adv. Nonlinear Stud., 16 (2016), 821–842. https://doi.org/10.1515/ans-2015-5055 doi: 10.1515/ans-2015-5055
[10]	M. del Pino, P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var., 4 (1996), 121–137. https://doi.org/10.1007/BF01189950 doi: 10.1007/BF01189950
[11]	Y. Ding, F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var., 30 (2007), 231–249. https://doi.org/10.1007/s00526-007-0091-z doi: 10.1007/s00526-007-0091-z
[12]	A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397–408. https://doi.org/10.1016/0022-1236(86)90096-0 doi: 10.1016/0022-1236(86)90096-0
[13]	M. M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937–1961. https://doi.org/10.1088/0951-7715/28/6/1937 doi: 10.1088/0951-7715/28/6/1937
[14]	C. F. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Part. Diff. Eq., 21 (1996), 787–820. https://doi.org/10.1080/03605309608821208 doi: 10.1080/03605309608821208
[15]	Y. He, G. Li, The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927–1952. https://doi.org/10.1007/s11425-014-4830-2 doi: 10.1007/s11425-014-4830-2
[16]	H. Jin, W. Liu, J. Zhang, Singularly perturbed fractional Schrödinger equation involving a general critical nonlinearity, Adv. Nonlinear Stud., 18 (2018), 487–499. https://doi.org/10.1515/ans-2018-2015 doi: 10.1515/ans-2018-2015
[17]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
[18]	N. Laskin, Fractional Schröding equation, Phys. Rev. E, 66 (2002), 056108. https://doi.org/10.1103/PhysRevE.66.056108 doi: 10.1103/PhysRevE.66.056108
[19]	Q. Lou, H. Luo, Multiplicity and concentration of positive solutions for fractional p-Laplacian problem involving concave-convex nonlinearity, Nonlinear Anal. Real, 42 (2018), 387–408. https://doi.org/10.1016/j.nonrwa.2018.01.013 doi: 10.1016/j.nonrwa.2018.01.013
[20]	G. Lu, Y. Shen, Existence of solutions to fractional p-Laplacian system with homogeneous nonlinearities of critical sobolev growth, Adv. Nonlinear Stud., 20 (2020), 579–597. https://doi.org/10.1515/ans-2020-2098 doi: 10.1515/ans-2020-2098
[21]	Q. Li, Z. Yang, Existence and multiplicity of solutions for perturbed fractional p-Laplacian equations with critical nonlinearity in $\mathbb{R}^{N}$, Appl. Anal., in press. https://doi.org/10.1080/00036811.2022.2045969
[22]	A. Mellet, S. Mischler, C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rational Mech. Anal., 199 (2011), 493–525. https://doi.org/10.1007/s00205-010-0354-2 doi: 10.1007/s00205-010-0354-2
[23]	J. M. do Ó, On existence and concentration of positive bound states of p-Laplacian equations in $\mathbb{R}^{N}$ involving critical growth, Nonlinear Anal. Theor., 62 (2005), 777–801. https://doi.org/10.1016/j.na.2005.03.093 doi: 10.1016/j.na.2005.03.093
[24]	P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$, Calc. Var., 54 (2015), 2785–2806. https://doi.org/10.1007/s00526-015-0883-5 doi: 10.1007/s00526-015-0883-5
[25]	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
[26]	P. H. Rabinowitz, On a class of nonlinear Schröding equations, Z. Angew. Math. Phys., 43 (1992), 270–291. https://doi.org/10.1007/BF00946631 doi: 10.1007/BF00946631
[27]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67–112. https://doi.org/10.1002/cpa.20153 doi: 10.1002/cpa.20153
[28]	B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Annali di Matematica, 181 (2002), 73–83. https://doi.org/10.1007/s102310200029 doi: 10.1007/s102310200029
[29]	Y. Song, S. Shi, Existence and multiplicity solutions for the p-Fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Complex Var. Elliptic Equ., 64 (2019), 1163–1183. https://doi.org/10.1080/17476933.2018.1511707 doi: 10.1080/17476933.2018.1511707
[30]	Y. Shen, Existence of solutions to elliptic problems with fractional p-Laplacian and multiple critical nonlinearities in the entire space, Nonlinear Anal., 202 (2021), 112102. https://doi.org/10.1016/j.na.2020.112102 doi: 10.1016/j.na.2020.112102
[31]	W. Wang, X. Yang, F. Zhao, Existence and concentration of ground state solutions for a subcubic quasilinear problem via Pohozaev manifold, J. Math. Anal. Appl., 424 (2015), 1471–1490. https://doi.org/10.1016/j.jmaa.2014.12.013 doi: 10.1016/j.jmaa.2014.12.013
[32]	L. Wang, X. Fang, Z. Han, Existence of standing wave solutions for coupled quasilinear Schrodinger systems with critical exponents in $\mathbb{R}^{N}$, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 12. https://doi.org/10.14232/ejqtde.2017.1.12 doi: 10.14232/ejqtde.2017.1.12
[33]	J. Wang, The existence of semiclassical states for some p-Laplacian equation with critical exponent, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 417–434. https://doi.org/10.1007/s10255-017-0671-4 doi: 10.1007/s10255-017-0671-4
[34]	M. Xiang, 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
[35]	M. Xiang, B. Zhang, X. Zhang, A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in $\mathbb{R}^{N}$, Adv. Nonlinear Stud., 17 (2017), 611–640. https://doi.org/10.1515/ans-2016-6002 doi: 10.1515/ans-2016-6002
[36]	M. Xiang, B. Zhang, Z. Wei, Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional p-Laplacian, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 107. https://doi.org/10.14232/ejqtde.2016.1.107 doi: 10.14232/ejqtde.2016.1.107
[37]	J. Xie, X. Huang, Y. Chen, Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term, Acta Math. Sci., 38 (2018), 1821–1832. https://doi.org/10.1016/S0252-9602(18)30849-X doi: 10.1016/S0252-9602(18)30849-X
[38]	M. Xiang, B. Zhang, V. Rǎdulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv Nonlinear Anal., 9 (2020), 690–709. https://doi.org/10.1515/anona-2020-0021 doi: 10.1515/anona-2020-0021
[39]	M. Yang, Y. Ding, Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in $\mathbb{R}^{N}$, Annali di Matematica, 192 (2013), 783–804. https://doi.org/10.1007/s10231-011-0246-6 doi: 10.1007/s10231-011-0246-6
[40]	H. Zhang, W. Liu, Existence of nontrivial solutions to perturbed p-Laplacian system in $\mathbb{R}^{N}$ involving critical nonlinearity, Bound. Value Probl., 2012 (2012), 53. https://doi.org/10.1186/1687-2770-2012-53 doi: 10.1186/1687-2770-2012-53
[41]	B. Zhang, M. Squassina, X. Zhang, Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscripta Math., 155 (2018), 115–140. https://doi.org/10.1007/s00229-017-0937-4 doi: 10.1007/s00229-017-0937-4
[42]	M. Zhen, B. Zhang, The Nehari manifold for fractional p-Laplacian system involving concave-conve nonlinearities and sign-changing weight functions, Complex Var. Elliptic Equ., 66 (2021), 1731–1754. https://doi.org/10.1080/17476933.2020.1779237 doi: 10.1080/17476933.2020.1779237
[43]	W. Zhang, S. Yuan, L. Wen, Existence and concentration of ground states for fractional Choquard equation with indefinite potential, Adv. Nonlinear Anal., 11 (2022), 1552–1578. https://doi.org/10.1515/anona-2022-0255 doi: 10.1515/anona-2022-0255
[44]	J. Zhang, W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), 114. https://doi.org/10.1007/s12220-022-00870-x doi: 10.1007/s12220-022-00870-x
[45]	W. Zhang, J. Zhang, H. Mi, Ground states and multiple solutions for Hamiltonian elliptic system with gradient term, Adv. Nonlinear Anal., 10 (2021), 331–352. https://doi.org/10.1515/anona-2020-0113 doi: 10.1515/anona-2020-0113
[46]	W. Zhang, J. Zhang, Multiplicity and concentration of positive solutions for fractional unbalanced double-phase problems, J. Geom. Anal., 32 (2022), 235. https://doi.org/10.1007/s12220-022-00983-3 doi: 10.1007/s12220-022-00983-3

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)