# American Institute of Mathematical Sciences

April  2022, 27(4): 1927-1954. doi: 10.3934/dcdsb.2021115

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

 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 Published  April 2022 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 , $ 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
,
 $a_{i},b_{i}\geq 0,$
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.
Citation: Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1927-1954. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [29] M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [29] M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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. [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. [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.
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