# American Institute of Mathematical Sciences

June  2018, 11(3): 533-545. doi: 10.3934/dcdss.2018029

## Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity

 1 College of Science, China University of Mining and Technology, Xuzhou 221116, China 2 College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

* Corresponding author: W. Liu.

Received  May 2017 Revised  August 2017 Published  October 2017

In this paper, we are concerned with the following fractional Kirchhoff equation
 \begin{align*}\left\{\begin{array}{ll}\left(a+b∈t_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2\right)(-Δ)^su=\lambda u+μ|u|^{q-2}u+|u|^{2_{s}^*-2}u&\ \ \mbox{in}\ \ Ω, \\ u=0&\ \ \mbox{in}\ \mathbb R^N\backslashΩ, \end{array}\right.\end{align*}
where
 $N>2s$
,
 $a, b, \lambda, μ>0$
,
 $s∈(0, 1)$
and
 $Ω$
is a bounded open domain with continuous boundary. Here
 $(-Δ)^s$
is the fractional Laplacian operator. For
 $2 , we prove that if $b$is small or $μ$is large, the problem above admits multiple solutions by virtue of a linking theorem due to G. Cerami, D. Fortunato and M. Struwe [7, Theorem 2.5]. Citation: Hua Jin, Wenbin Liu, Jianjun Zhang. Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 533-545. doi: 10.3934/dcdss.2018029 ##### References:  [1] V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in$\mathbb R^N$with a general nonlinearity Commun. Contemp. Math. published online (2017). doi: 10.1142/S0219199717500547. Google Scholar [2] D. Applebaum, Lévy processes-from probability theory to finance and quantum groups, Notices of the American Math Soc., 51 (2004), 1336-1347. Google Scholar [3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems invoving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [4] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincare Anal. Nonlineaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003. Google Scholar [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorem and applications to some nonlinear problems with ''strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3. Google Scholar [6] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Apple. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [7] G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 341-350. doi: 10.1016/S0294-1449(16)30416-4. Google Scholar [8] Z. Chen, N. Shioji and W. Zou, Ground state and multiple solutions for a critical exponent problem, Nonlinear Differ. Equ. Appl., 19 (2012), 253-277. doi: 10.1007/s00030-011-0127-0. Google Scholar [9] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in$\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. Google Scholar [10] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 512-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [11] A. Fiscella, G. M. Bisci and R. Servadei, Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems, Bull. Sci. Math., 140 (2016), 14-35. doi: 10.1016/j.bulsci.2015.10.001. Google Scholar [12] A. Fiscella, R. Servadei and E. Valdinoci, Denisity properties for fractional Sobolev space, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar [13] 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 [14] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in$\mathbb R^3$involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510. doi: 10.1515/ans-2014-0214. Google Scholar [15] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in$\mathbb R^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar [16] L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schröinger equations, Adv. Differential Equations, 11 (2006), 813-840. Google Scholar [17] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883. Google Scholar [18] N. Laskin, Fractional quantum mechanics and Levy path integrals, Physics Letters A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [19] Y. Liu, Z. Liu and Z. Ouyang, Multiplicity results for the Kirchhlff type equations with critical growth, Appl. Math. Lett., 63 (2017), 118-123. doi: 10.1016/j.aml.2016.07.029. Google Scholar [20] Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension Nonlinear Differ. Equ. Appl. 24 (2017), Art. 50, 32 pp. doi: 10.1007/s00030-017-0473-7. Google Scholar [21] D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, Nonlinear Differ. Equ. Appl., 21 (2014), 885-914. doi: 10.1007/s00030-014-0271-4. Google Scholar [22] D. Naimen, On the Brezis-Nirenberg problem with a Kirchhoff Type Perturbation, Adv. Nonlinear Stud., 15 (2015), 135-156. doi: 10.1515/ans-2015-0107. Google Scholar [23] G. Palatucci and E. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc, Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y. Google Scholar [24] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006. Google Scholar [25] 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 [26] M. Schether and W. Zou, On the Brezis-Nirenberg problem, Arch. Ration. Mech. Anal., 197 (2010), 337-356. doi: 10.1007/s00205-009-0288-8. Google Scholar [27] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2015-2137. Google Scholar [28] 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 [29] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar [30] R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut., 28 (2015), 655-676. doi: 10.1007/s13163-015-0170-1. Google Scholar [31] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [32] Y. Sire and E. Valdinoci, Fractional Laplacian phase transtion and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [33] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. Google Scholar [34] M. Xiang, P. Pucci, M. Squassina and B. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst., 37 (2017), 1631-1649. doi: 10.3934/dcds.2017067. Google Scholar [35] M. Xiang, B. Zhang and M. Yang, A fractional Kirchhoff-type problem in$\mathbb R^N$without the (AR) condition, Complex Var. Elliptic Equ., 61 (2016), 1481-1493. doi: 10.1080/17476933.2016.1182519. Google Scholar show all references ##### References:  [1] V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in$\mathbb R^N$with a general nonlinearity Commun. Contemp. Math. published online (2017). doi: 10.1142/S0219199717500547. Google Scholar [2] D. Applebaum, Lévy processes-from probability theory to finance and quantum groups, Notices of the American Math Soc., 51 (2004), 1336-1347. Google Scholar [3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems invoving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [4] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincare Anal. Nonlineaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003. Google Scholar [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorem and applications to some nonlinear problems with ''strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3. Google Scholar [6] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Apple. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [7] G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 341-350. doi: 10.1016/S0294-1449(16)30416-4. Google Scholar [8] Z. Chen, N. Shioji and W. Zou, Ground state and multiple solutions for a critical exponent problem, Nonlinear Differ. Equ. Appl., 19 (2012), 253-277. doi: 10.1007/s00030-011-0127-0. Google Scholar [9] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in$\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. Google Scholar [10] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 512-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [11] A. Fiscella, G. M. Bisci and R. Servadei, Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems, Bull. Sci. Math., 140 (2016), 14-35. doi: 10.1016/j.bulsci.2015.10.001. Google Scholar [12] A. Fiscella, R. Servadei and E. Valdinoci, Denisity properties for fractional Sobolev space, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar [13] 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 [14] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in$\mathbb R^3$involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510. doi: 10.1515/ans-2014-0214. Google Scholar [15] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in$\mathbb R^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar [16] L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schröinger equations, Adv. Differential Equations, 11 (2006), 813-840. Google Scholar [17] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883. Google Scholar [18] N. Laskin, Fractional quantum mechanics and Levy path integrals, Physics Letters A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [19] Y. Liu, Z. Liu and Z. Ouyang, Multiplicity results for the Kirchhlff type equations with critical growth, Appl. Math. Lett., 63 (2017), 118-123. doi: 10.1016/j.aml.2016.07.029. Google Scholar [20] Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension Nonlinear Differ. Equ. Appl. 24 (2017), Art. 50, 32 pp. doi: 10.1007/s00030-017-0473-7. Google Scholar [21] D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, Nonlinear Differ. Equ. Appl., 21 (2014), 885-914. doi: 10.1007/s00030-014-0271-4. Google Scholar [22] D. Naimen, On the Brezis-Nirenberg problem with a Kirchhoff Type Perturbation, Adv. Nonlinear Stud., 15 (2015), 135-156. doi: 10.1515/ans-2015-0107. Google Scholar [23] G. Palatucci and E. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc, Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y. Google Scholar [24] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006. Google Scholar [25] 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 [26] M. Schether and W. Zou, On the Brezis-Nirenberg problem, Arch. Ration. Mech. Anal., 197 (2010), 337-356. doi: 10.1007/s00205-009-0288-8. Google Scholar [27] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2015-2137. Google Scholar [28] 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 [29] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445. Google Scholar [30] R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut., 28 (2015), 655-676. doi: 10.1007/s13163-015-0170-1. Google Scholar [31] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [32] Y. Sire and E. Valdinoci, Fractional Laplacian phase transtion and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [33] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. Google Scholar [34] M. Xiang, P. Pucci, M. Squassina and B. 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