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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.

[2]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645.

[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.

[4]

G. AutuoriA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

E. Di NezzaG. 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]

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.

[14]

P. FelmerA. 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.

[15]

G. M. FigueiredoN. 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.

[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.

[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.

[18]

F. Gazzola and M. Lazzarino, Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50.

[19]

G. GuW. 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.

[20]

G. GuW. 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.

[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.

[22]

Y. HeG. 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.

[23]

G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088. doi: 10.1142/S0219199715500881.

[31]

P. PucciM. 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.

[32]

P. PucciM. 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.

[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.

[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.

[35]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.

[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.

[37]

A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010.

[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.

[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.

[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.

[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.

[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.

[43]

J. ZhangZ. LouY. 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.

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.

[2]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645.

[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.

[4]

G. AutuoriA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

E. Di NezzaG. 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]

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.

[14]

P. FelmerA. 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.

[15]

G. M. FigueiredoN. 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.

[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.

[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.

[18]

F. Gazzola and M. Lazzarino, Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50.

[19]

G. GuW. 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.

[20]

G. GuW. 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.

[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.

[22]

Y. HeG. 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.

[23]

G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088. doi: 10.1142/S0219199715500881.

[31]

P. PucciM. 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.

[32]

P. PucciM. 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.

[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.

[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.

[35]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.

[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.

[37]

A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010.

[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.

[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.

[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.

[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.

[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.

[43]

J. ZhangZ. LouY. 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.

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