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A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents

Maoding Zhen 1, , Binlin Zhang 2,, and  Xiumei Han 3,

1. 

School of Mathematics, Hefei University of Technology, Hefei, 230009, China
2. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China
3. 

College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao, 266590, China

* Corresponding author: Binlin Zhang

Received  February 2021 Early access April 2021

In this paper, we study the following Kirchhoff-type fractional Schrödinger system with critical exponent in
$ \mathbb{R}^N $
:
$ \begin{equation*} \begin{cases} \left(a_{1}+b_{1}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u +\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u,\\ \left(a_{2}+b_{2}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx\right)(-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+ \frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\\ \end{cases} \end{equation*} $
where
$ (-\Delta)^{s} $
 is the fractional Laplacian,
$ 0<s<1 $
,
$ N>2s, $
$ 2_{s}^{\ast} = 2N/(N-2s) $
 is the fractional critical Sobolev exponent,
$ \mu_{1},\mu_{2},\gamma, k>0 $
,
$ \alpha+\beta = 2_{s}^{\ast},\ 1<p<2_{s}^{\ast}-1 $
,
$ a_{i},b_{i}\geq 0, $
 with
$ a_{i}+b_{i}>0,\ \ i = 1,2 $
. By using appropriate transformation, we first get its equivalent system which may be easier to solve:
$ \begin{equation*} \begin{cases} (-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u+\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u, \ \ x\in \mathbb{R}^N, \\ (-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+\frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\ \ x\in \mathbb{R}^N,\\ \lambda_{1}^{s}-a_{1}-b_{1}\lambda_{1}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx = 0, \ \ \lambda_{1}\in \mathbb{R}^+,\\ \lambda_{2}^{s}-a_{2}-b_{2}\lambda_{2}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx = 0, \ \ \lambda_{2}\in \mathbb{R}^+. \end{cases} \end{equation*} $
Then, by using the mountain pass theorem, together with some classical arguments from Brézis and Nirenberg, we obtain the existence of solutions for the new system under suitable conditions. Finally, based on the equivalence of two systems, we get the existence of solutions for the original system. Our results give improvement and complement of some recent theorems in several directions.
Keywords: Kirchhoff-type problem, Fractional Schrödinger system, Critical exponent, Mountain pass theorem.
Mathematics Subject Classification: Primary: 35J20, 35B33, 58E05.
Citation: Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021115
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

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V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity, Commun. Contemp. Math., 20 (2018), 1750054, 17pp. doi: 10.1142/S0219199717500547.  Google Scholar

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H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure. Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

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S. Baraket and G. Molica Bisci, Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal., 6 (2017), 85-93.  doi: 10.1515/anona-2015-0168.  Google Scholar

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G. M. Bisci, Sequence of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 241-253.  doi: 10.4310/MRL.2014.v21.n2.a3.  Google Scholar

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H. Brézis and E. Lieb, A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

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Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.  Google Scholar

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Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.  Google Scholar

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A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar

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P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent Math, 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar

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Y. Ding, F. Gao and M. Yang, Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.  doi: 10.1088/1361-6544/aba88d.  Google Scholar

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[15]

A. Fiscella and P. Pucci, $p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004.  Google Scholar

[16]

F. Gao, E. Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.  Google Scholar

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Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069.  Google Scholar

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X. He and W. Zou, Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.  doi: 10.1007/s11425-017-9399-6.  Google Scholar

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P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257.   Google Scholar

[20]

Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 32 pp. doi: 10.1007/s00030-017-0473-7.  Google Scholar

[21]

D. Lü and S. Peng, Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differential Equation, 263 (2017), 8947-8978.  doi: 10.1016/j.jde.2017.08.062.  Google Scholar

[22] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
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A. Mellet, S. Mischler and C. Mouhotg, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[24]

P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differetial Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[25]

P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[26]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[27]

R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[29]

M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[30]

K. Wu and F. Zhou, Nodal solutions for a Kirchhoff type problem in $\mathbb{R}^N$, Appl. Math. Lett., 88 (2019), 58-63.  doi: 10.1016/j.aml.2018.08.008.  Google Scholar

[31]

M. Xiang, B. Zhang and V. Rădulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional $p$-Laplacian and critical exponent, Adv. Nonlinear Anal., 9 (2020), 690-709.  doi: 10.1515/anona-2020-0021.  Google Scholar

[32]

M. Zhen, J. He, H. Xu and M. Yang, Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent, Discrete Contin. Dyn. Syst., 39 (2019), 6523-6539.  doi: 10.3934/dcds.2019283.  Google Scholar

[33]

M. Zhen, B. Zhang and V. Rădulescu, Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653-2676.  doi: 10.3934/dcds.2020379.  Google Scholar

[34]

F. Zhou and M. Yang, Solutions for a Kirchhoff type problem with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 494 (2021), 124638, 7pp. doi: 10.1016/j. jmaa. 2020.124638.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity, Commun. Contemp. Math., 20 (2018), 1750054, 17pp. doi: 10.1142/S0219199717500547.  Google Scholar

[3]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure. Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[4]

S. Baraket and G. Molica Bisci, Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal., 6 (2017), 85-93.  doi: 10.1515/anona-2015-0168.  Google Scholar

[5]

G. M. Bisci, Sequence of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 241-253.  doi: 10.4310/MRL.2014.v21.n2.a3.  Google Scholar

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[7]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.  Google Scholar

[8]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.  Google Scholar

[9]

A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar

[10]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent Math, 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar

[11]

Y. Ding, F. Gao and M. Yang, Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.  doi: 10.1088/1361-6544/aba88d.  Google Scholar

[12]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[13]

L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847-5866.  doi: 10.3934/dcds.2019219.  Google Scholar

[14]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[15]

A. Fiscella and P. Pucci, $p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004.  Google Scholar

[16]

F. Gao, E. Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.  Google Scholar

[17]

Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069.  Google Scholar

[18]

X. He and W. Zou, Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.  doi: 10.1007/s11425-017-9399-6.  Google Scholar

[19]

P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257.   Google Scholar

[20]

Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 32 pp. doi: 10.1007/s00030-017-0473-7.  Google Scholar

[21]

D. Lü and S. Peng, Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differential Equation, 263 (2017), 8947-8978.  doi: 10.1016/j.jde.2017.08.062.  Google Scholar

[22] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[23]

A. Mellet, S. Mischler and C. Mouhotg, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[24]

P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differetial Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[25]

P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[26]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[27]

R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[29]

M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[30]

K. Wu and F. Zhou, Nodal solutions for a Kirchhoff type problem in $\mathbb{R}^N$, Appl. Math. Lett., 88 (2019), 58-63.  doi: 10.1016/j.aml.2018.08.008.  Google Scholar

[31]

M. Xiang, B. Zhang and V. Rădulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional $p$-Laplacian and critical exponent, Adv. Nonlinear Anal., 9 (2020), 690-709.  doi: 10.1515/anona-2020-0021.  Google Scholar

[32]

M. Zhen, J. He, H. Xu and M. Yang, Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent, Discrete Contin. Dyn. Syst., 39 (2019), 6523-6539.  doi: 10.3934/dcds.2019283.  Google Scholar

[33]

M. Zhen, B. Zhang and V. Rădulescu, Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653-2676.  doi: 10.3934/dcds.2020379.  Google Scholar

[34]

F. Zhou and M. Yang, Solutions for a Kirchhoff type problem with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 494 (2021), 124638, 7pp. doi: 10.1016/j. jmaa. 2020.124638.  Google Scholar

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