November  2019, 12(7): 1929-1954. doi: 10.3934/dcdss.2019126

Least energy solutions for fractional Kirchhoff type equations involving critical growth

1. 

School of Mathematics and information, Guangxi University, Nanning 530004, China

2. 

Department of Mathematics, Central China Normal University, Wuhan 430079, China

3. 

School of Science, East China JiaoTong University, Nanchang 330013, China

* Corresponding author: Yinbin Deng

Received  November 2017 Revised  April 2018 Published  December 2018

We study the following fractional Kirchhoff type equation:
$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $
where
$ a, \ b>0 $
are constants,
$ 2^*_s = \frac{6}{3-2s} $
with
$ s\in(0, 1) $
is the critical Sobolev exponent in
$ \mathbb{R} ^3 $
,
$ V $
is a potential function on
$ \mathbb{R} ^3 $
. Under some more general assumptions on
$ f $
and
$ V $
, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.
Citation: Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
References:
[1]

C. O. AlvesF. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Am. Math. Soc., 51 (2004), 1336-1347. 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]

P. BilerG. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4. Google Scholar

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C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. Google Scholar

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L. CaffarelliJ. M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331. Google Scholar

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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]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6. Google Scholar

[9]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. Google Scholar

[10]

X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494. doi: 10.1088/0951-7715/26/2/479. Google Scholar

[11]

C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. Google Scholar

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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. 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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S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $ \mathbb{R} ^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017.Google Scholar

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S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche (Catania), 68 (2013), 201-216. Google Scholar

[19]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976. Google Scholar

[20]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar

[21]

G. M. FigueiredoG. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361. doi: 10.3233/ASY-151316. Google Scholar

[22]

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

[23]

Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902. doi: 10.1016/j.jde.2015.04.005. Google Scholar

[24]

X. He and W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500. doi: 10.1007/s10231-012-0286-6. Google Scholar

[25]

X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021. Google Scholar

[26]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $ \mathbb{R} ^3$, . Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[27]

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

[28]

Y. HeG. 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

[29]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar

[30]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108. Google Scholar

[33]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $ \mathbb{R} ^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[34]

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), Art. 50, 32 pp. doi: 10.1007/s00030-017-0473-7. Google Scholar

[35]

E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312. doi: 10.1016/j.aim.2007.08.009. Google Scholar

[36]

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

[37]

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. Google Scholar

[38]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990. Google Scholar

[39]

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

[40]

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

[41]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187. Google Scholar

[42]

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

[43]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. Google Scholar

[44]

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), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. Google Scholar

[45]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2017), 3061-3106. doi: 10.1016/j.jde.2016.05.022. Google Scholar

[46]

J. WangL. TianJ. 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

[47]

M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar

[48]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ \mathbb{R} ^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

show all references

References:
[1]

C. O. AlvesF. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Am. Math. Soc., 51 (2004), 1336-1347. 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]

P. BilerG. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4. Google Scholar

[5]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. Google Scholar

[6]

L. CaffarelliJ. M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331. 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]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6. Google Scholar

[9]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. Google Scholar

[10]

X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494. doi: 10.1088/0951-7715/26/2/479. Google Scholar

[11]

C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar

[13]

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

[14]

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

[15]

Y. DengS. Peng and W. Shuai, Existence and asympototic 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

[16]

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. Google Scholar

[17]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $ \mathbb{R} ^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017.Google Scholar

[18]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche (Catania), 68 (2013), 201-216. Google Scholar

[19]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin, 1976. Google Scholar

[20]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar

[21]

G. M. FigueiredoG. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361. doi: 10.3233/ASY-151316. Google Scholar

[22]

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

[23]

Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902. doi: 10.1016/j.jde.2015.04.005. Google Scholar

[24]

X. He and W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500. doi: 10.1007/s10231-012-0286-6. Google Scholar

[25]

X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021. Google Scholar

[26]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $ \mathbb{R} ^3$, . Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[27]

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

[28]

Y. HeG. 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

[29]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar

[30]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56-108. doi: 10.1103/PhysRevE.66.056108. Google Scholar

[33]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $ \mathbb{R} ^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[34]

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), Art. 50, 32 pp. doi: 10.1007/s00030-017-0473-7. Google Scholar

[35]

E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312. doi: 10.1016/j.aim.2007.08.009. Google Scholar

[36]

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

[37]

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. Google Scholar

[38]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990. Google Scholar

[39]

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

[40]

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

[41]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187. Google Scholar

[42]

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

[43]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. Google Scholar

[44]

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), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. Google Scholar

[45]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2017), 3061-3106. doi: 10.1016/j.jde.2016.05.022. Google Scholar

[46]

J. WangL. TianJ. 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

[47]

M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar

[48]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ \mathbb{R} ^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

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