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November  2019, 18(6): 3181-3200. doi: 10.3934/cpaa.2019143

## Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent

 1 School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China 2 Department of Mathematics, University of Texas at San Antonio, San Antonio, 78249 Texas, USA

* Corresponding author

Received  November 2018 Revised  February 2019 Published  May 2019

Fund Project: The first author is supported by China Scholarship Council (201806370022), Hunan Provincial Innovation Foundation for Postgraduate (CX2018B052). The second author is supported by National Natural Science Foundation of China (11571370)

In this paper, we study the following fractional Kirchhoff equation with critical nonlinearity
 $\Big(a+b\int_{\mathbb{R}^3}| (-\Delta)^{\frac{s}{2}} u|^2dx\Big) (-\Delta )^su+V(x) u = K(x)|u|^{2_s^*-2}u+\lambda g(x,u), \; \text{in}\; \mathbb{R}^3,$
where
 $a,b>0$
,
 $\lambda>0$
,
 $(-\Delta )^s$
is the fractional Laplace operator with
 $s\in(\frac{3}{4},1)$
and
 $2_s^* = \frac{6}{3-2s}$
,
 $V,K$
and
 $g$
are asymptotically periodic in
 $x$
. The existence of a positive ground state solution is obtained by variational method.
Citation: Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
##### References:
 [1] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic kirchhoff equation in $\mathbb{R}^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. Google Scholar [2] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645. Google Scholar [3] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar [4] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [5] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93. doi: 10.1007/BF00375686. Google Scholar [6] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana doi: 10.1007/978-3-319-28739-3. Google Scholar [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [8] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. (4), 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x. Google Scholar [9] S. Chen and X. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar [10] S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 40, 23. doi: 10.1007/s00030-018-0531-9. Google Scholar [11] S. Chen and X. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. 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] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$ doi: 10.1007/978-88-7642-601-8. Google Scholar [14] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [15] G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar [16] G. M. Figueiredo and J. R. Santos Júnior, Existence of a least energy nodal solution for a schrödinger-kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506. doi: 10.1063/1.4921639. Google Scholar [17] 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 [18] F. Gazzola and M. Lazzarino, Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50. Google Scholar [19] G. Gu, W. Zhang and F. Zhao, Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55. doi: 10.1016/j.aml.2018.04.010. Google Scholar [20] G. Gu, W. Zhang and F. Zhao, Infinitely many sign-changing solutions for a nonlocal problem, Ann. Mat. Pura Appl., 197 (2018), 1429-14448. doi: 10.1007/s10231-018-0731-2. Google Scholar [21] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. Google Scholar [22] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\Bbb R^3$ involving critical Sobolev exponents, Adv. Nonlinear Studies, 14 (2014), 483-510. doi: 10.1515/ans-2014-0214. Google Scholar [23] G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883.Google Scholar [24] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar [25] Q. Li, K. Teng and X. Wu, Ground states for fractional schrödinger equations with critical growth, J. Math. Phys., 59 (2018), 033504. doi: 10.1063/1.5008662. Google Scholar [26] S. Liang and J. Zhang, Multiplicity of solutions for the noncooperative schrödinger-kirchhoff system involving the fractional p-laplacian in $\mathbb{R}^N$, Z. Angew. Math. Phys., 68 (2017), 63. doi: 10.1007/s00033-017-0805-9. Google Scholar [27] J. L. Lions,, On Some Questions in Boundary Value Problems of Mathematical Physics, volume 30 of North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978. Google Scholar [28] 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), no.4, Art. 50, 32. doi: 10.1007/s00030-017-0473-7. Google Scholar [29] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, volume 162 of Encyclopedia of Mathematics and its Applications. doi: 10.1017/CBO9781316282397. Google Scholar [30] G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088. doi: 10.1142/S0219199715500881. Google Scholar [31] 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 Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5. Google Scholar [32] 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 [33] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar [34] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105. Google Scholar [35] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. Google Scholar [36] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 36 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [37] A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010. Google Scholar [38] X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110 pp.1-25. doi: 10.1007/s00526-017-1214-9. Google Scholar [39] X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar [40] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. doi: 10.1016/j.jde.2016.05.022. Google Scholar [41] M. Willem, Minimax Theorems, volume 24 of Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [42] H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 56 (2015), 091502. doi: 10.1063/1.4929660. Google Scholar [43] J. Zhang, Z. Lou, Y. Ji and W. Shao, Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 462 (2018), 57-83. doi: 10.1016/j.jmaa.2018.01.060. Google Scholar

show all references

##### References:
 [1] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic kirchhoff equation in $\mathbb{R}^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. Google Scholar [2] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645. Google Scholar [3] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar [4] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [5] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93. doi: 10.1007/BF00375686. Google Scholar [6] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana doi: 10.1007/978-3-319-28739-3. Google Scholar [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [8] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. (4), 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x. Google Scholar [9] S. Chen and X. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar [10] S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 40, 23. doi: 10.1007/s00030-018-0531-9. Google Scholar [11] S. Chen and X. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. 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] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$ doi: 10.1007/978-88-7642-601-8. Google Scholar [14] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [15] G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar [16] G. M. Figueiredo and J. R. Santos Júnior, Existence of a least energy nodal solution for a schrödinger-kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506. doi: 10.1063/1.4921639. Google Scholar [17] 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 [18] F. Gazzola and M. Lazzarino, Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50. Google Scholar [19] G. Gu, W. Zhang and F. Zhao, Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55. doi: 10.1016/j.aml.2018.04.010. Google Scholar [20] G. Gu, W. Zhang and F. Zhao, Infinitely many sign-changing solutions for a nonlocal problem, Ann. Mat. Pura Appl., 197 (2018), 1429-14448. doi: 10.1007/s10231-018-0731-2. Google Scholar [21] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. Google Scholar [22] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\Bbb R^3$ involving critical Sobolev exponents, Adv. Nonlinear Studies, 14 (2014), 483-510. doi: 10.1515/ans-2014-0214. Google Scholar [23] G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883.Google Scholar [24] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar [25] Q. Li, K. Teng and X. Wu, Ground states for fractional schrödinger equations with critical growth, J. Math. Phys., 59 (2018), 033504. doi: 10.1063/1.5008662. Google Scholar [26] S. Liang and J. Zhang, Multiplicity of solutions for the noncooperative schrödinger-kirchhoff system involving the fractional p-laplacian in $\mathbb{R}^N$, Z. Angew. Math. Phys., 68 (2017), 63. doi: 10.1007/s00033-017-0805-9. Google Scholar [27] J. L. Lions,, On Some Questions in Boundary Value Problems of Mathematical Physics, volume 30 of North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978. Google Scholar [28] 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), no.4, Art. 50, 32. doi: 10.1007/s00030-017-0473-7. Google Scholar [29] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, volume 162 of Encyclopedia of Mathematics and its Applications. doi: 10.1017/CBO9781316282397. Google Scholar [30] G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088. doi: 10.1142/S0219199715500881. Google Scholar [31] 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 Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5. Google Scholar [32] 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 [33] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar [34] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105. Google Scholar [35] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. Google Scholar [36] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 36 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [37] A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010. Google Scholar [38] X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110 pp.1-25. doi: 10.1007/s00526-017-1214-9. Google Scholar [39] X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar [40] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. doi: 10.1016/j.jde.2016.05.022. Google Scholar [41] M. Willem, Minimax Theorems, volume 24 of Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [42] H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 56 (2015), 091502. doi: 10.1063/1.4929660. Google Scholar [43] J. Zhang, Z. Lou, Y. Ji and W. Shao, Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 462 (2018), 57-83. doi: 10.1016/j.jmaa.2018.01.060. Google Scholar
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