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January  2019, 18(1): 129-158. doi: 10.3934/cpaa.2019008

## Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, Guangdong, China

* Corresponding author

Received  September 2017 Revised  October 2017 Published  August 2018

Fund Project: The first author is supported by National Natural Science Foundation of China grant 11701113 and China Postdoctoral Science Foundation funded project grant 2016M600647. The second author is supported by National Natural Science Foundation of China grant 11471085, Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226 and Guangdong Innovative Research Team Program grant 2011S009.

In the present paper, we consider the following Kirchhoff type problem
 $\begin{cases}-\Big(a+λ∈t_{\mathbb R^N} | \nabla u|^2dx\Big) Δ u+V(x)u = |u|^{2^*-2}u \;\;\;{\rm in}\ \mathbb{R}^N,\\u∈ D^{1,2}(\mathbb R^N),\end{cases}$
where
 $a$
is a positive constant,
 $λ$
is a positive parameter,
 $V∈ L^{\frac{N}{2}}(\mathbb{R}^N)$
is a given nonnegative function and
 $2^*$
is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole
 $\mathbb R^N$
(
 $N≥5$
) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension
 $N≥4$
, and especially it is the fist time to consider the case when
 $N≥5$
in the literature.
Citation: Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure & Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008
##### References:
  C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2 (2010), 409-417.  doi: 10.7153/dea-02-25.  Google Scholar  V. Benci and G. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u = u^{(N+2)/(N-2)}$ in $\mathbb R^N$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar  D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.  Google Scholar  P. Chen and X. Liu, Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 17 (2018), 113-125.  doi: 10.3934/cpaa.2018007.  Google Scholar  W. Y. Ding, On a Conformally Invariant Elliptic Equation on $\mathbb R^N$, Commun. Math. Phys., 107 (1986), 331-335. Google Scholar  P. L. Felmer, A. Quaas, M. X. Tang and J. S. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. I. H. Poincaré-AN., 25 (2008), 105-119.  doi: 10.1016/j.anihpc.2006.12.003.  Google Scholar  G. M. Figueiredo, R. C. Morales, J. J. R. Santos and A. Suárez, Study of a nonlinear Kirchhoff equation with non-homogeneous material, J. Math. Anal. Appl., 416 (2014), 597-608.  doi: 10.1016/j.jmaa.2014.02.067.  Google Scholar  X. M. He and W. M. 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  Y. S. Huang, Z. Liu and Y. Z. Wu, On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems, Mathematics, 7 (2015), 97-114.   Google Scholar  Y. S. Huang, Z. Liu and Y. Z. Wu, On finding solutions of a Kirchhoff type problem, Proc. Amer. Math. Soc., 144 (2016), 3019-3033.  doi: 10.1090/proc/12946.  Google Scholar  G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb R^3$ with critical Sobolev exponent, Math. Methods Appl. Sci., 37 (2014), 2570-2584.  doi: 10.1002/mma.3000.  Google Scholar  Z. S. Liu and S. J. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl., 426 (2015), 267-287.  doi: 10.1016/j.jmaa.2015.01.044.  Google Scholar  Z. S. Liu and S. J. Guo, Existence and concentration of positive ground state for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.  doi: 10.1007/s00033-014-0431-8.  Google Scholar  J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb R^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066.  Google Scholar  Z. Liu, S. Guo and Y. Fang, Positive solutions of Kirchhoff type elliptic equations in $\mathbb R^4$ with critical growth, Mathematische Nachrichten, 290 (2017), 367-381.  doi: 10.1002/mana.201500358.  Google Scholar  R. Q. Liu, C. L. Tang, J. F. Liao and X. P. Wu, Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four, Commun. Pure Appl. Anal., 15 (2016), 1841-1856.  doi: 10.3934/cpaa.2016006.  Google Scholar  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  D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar  G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. I. H. Poincaré-AN., 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar  M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar  J. Wang, L. X. Tian, J. X. Xu and F. B. 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  J. Wang and L. Xiao, Existence and concentration of solutions for a Kirchhoff tpye problem with potentials, Discrete Contin. Dyn. Syst. A, 12 (2016), 7137-7168.  doi: 10.3934/dcds.2016111.  Google Scholar  Y. J. Sun and X. Liu, Existence of positive solutions for Kirchhoff type problems with critical exponent, J. Partial Differ. Equ., 25 (2012), 187-198. Google Scholar  Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786.  doi: 10.3934/cpaa.2013.12.2773.  Google Scholar  Q. L. Xie, S. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028.  Google Scholar

show all references

##### References:
  C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2 (2010), 409-417.  doi: 10.7153/dea-02-25.  Google Scholar  V. Benci and G. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u = u^{(N+2)/(N-2)}$ in $\mathbb R^N$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar  D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.  Google Scholar  P. Chen and X. Liu, Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 17 (2018), 113-125.  doi: 10.3934/cpaa.2018007.  Google Scholar  W. Y. Ding, On a Conformally Invariant Elliptic Equation on $\mathbb R^N$, Commun. Math. Phys., 107 (1986), 331-335. Google Scholar  P. L. Felmer, A. Quaas, M. X. Tang and J. S. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. I. H. Poincaré-AN., 25 (2008), 105-119.  doi: 10.1016/j.anihpc.2006.12.003.  Google Scholar  G. M. Figueiredo, R. C. Morales, J. J. R. Santos and A. Suárez, Study of a nonlinear Kirchhoff equation with non-homogeneous material, J. Math. Anal. Appl., 416 (2014), 597-608.  doi: 10.1016/j.jmaa.2014.02.067.  Google Scholar  X. M. He and W. M. 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  Y. S. Huang, Z. Liu and Y. Z. Wu, On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems, Mathematics, 7 (2015), 97-114.   Google Scholar  Y. S. Huang, Z. Liu and Y. Z. Wu, On finding solutions of a Kirchhoff type problem, Proc. Amer. Math. Soc., 144 (2016), 3019-3033.  doi: 10.1090/proc/12946.  Google Scholar  G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb R^3$ with critical Sobolev exponent, Math. Methods Appl. Sci., 37 (2014), 2570-2584.  doi: 10.1002/mma.3000.  Google Scholar  Z. S. Liu and S. J. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl., 426 (2015), 267-287.  doi: 10.1016/j.jmaa.2015.01.044.  Google Scholar  Z. S. Liu and S. J. Guo, Existence and concentration of positive ground state for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.  doi: 10.1007/s00033-014-0431-8.  Google Scholar  J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb R^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066.  Google Scholar  Z. Liu, S. Guo and Y. Fang, Positive solutions of Kirchhoff type elliptic equations in $\mathbb R^4$ with critical growth, Mathematische Nachrichten, 290 (2017), 367-381.  doi: 10.1002/mana.201500358.  Google Scholar  R. Q. Liu, C. L. Tang, J. F. Liao and X. P. Wu, Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four, Commun. Pure Appl. Anal., 15 (2016), 1841-1856.  doi: 10.3934/cpaa.2016006.  Google Scholar  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  D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar  G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. I. H. Poincaré-AN., 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar  M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar  J. Wang, L. X. Tian, J. X. Xu and F. B. 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  J. Wang and L. Xiao, Existence and concentration of solutions for a Kirchhoff tpye problem with potentials, Discrete Contin. Dyn. Syst. A, 12 (2016), 7137-7168.  doi: 10.3934/dcds.2016111.  Google Scholar  Y. J. Sun and X. Liu, Existence of positive solutions for Kirchhoff type problems with critical exponent, J. Partial Differ. Equ., 25 (2012), 187-198. Google Scholar  Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786.  doi: 10.3934/cpaa.2013.12.2773.  Google Scholar  Q. L. Xie, S. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028.  Google Scholar
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