• Previous Article
    Boundary regularity for a degenerate elliptic equation with mixed boundary conditions
  • CPAA Home
  • This Issue
  • Next Article
    Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity
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:
[1]

C. O. AlvesF. 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.

[2]

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.

[3]

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.

[4]

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.

[5]

W. Y. Ding, On a Conformally Invariant Elliptic Equation on $\mathbb R^N$, Commun. Math. Phys., 107 (1986), 331-335.

[6]

P. L. FelmerA. QuaasM. 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.

[7]

G. M. FigueiredoR. C. MoralesJ. 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.

[8]

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.

[9]

Y. S. HuangZ. Liu and Y. Z. Wu, On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems, Mathematics, 7 (2015), 97-114.

[10]

Y. S. HuangZ. 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.

[11]

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.

[12]

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.

[13]

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.

[14]

J. LiuJ. 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.

[15]

Z. LiuS. 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.

[16]

R. Q. LiuC. L. TangJ. 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.

[17]

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.

[18]

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.

[19]

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.

[20]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[21]

J. WangL. X. TianJ. 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.

[22]

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.

[23]

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.

[24]

Q. L. XieX. 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.

[25]

Q. L. XieS. 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.

show all references

References:
[1]

C. O. AlvesF. 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.

[2]

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.

[3]

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.

[4]

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.

[5]

W. Y. Ding, On a Conformally Invariant Elliptic Equation on $\mathbb R^N$, Commun. Math. Phys., 107 (1986), 331-335.

[6]

P. L. FelmerA. QuaasM. 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.

[7]

G. M. FigueiredoR. C. MoralesJ. 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.

[8]

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.

[9]

Y. S. HuangZ. Liu and Y. Z. Wu, On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems, Mathematics, 7 (2015), 97-114.

[10]

Y. S. HuangZ. 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.

[11]

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.

[12]

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.

[13]

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.

[14]

J. LiuJ. 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.

[15]

Z. LiuS. 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.

[16]

R. Q. LiuC. L. TangJ. 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.

[17]

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.

[18]

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.

[19]

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.

[20]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[21]

J. WangL. X. TianJ. 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.

[22]

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.

[23]

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.

[24]

Q. L. XieX. 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.

[25]

Q. L. XieS. 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.

Figure 3.  $N\geq5$
[1]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[2]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[3]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006

[4]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721

[5]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[6]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[7]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[8]

Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943

[9]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030

[10]

Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078

[11]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[12]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139

[13]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417

[14]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

[15]

Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711

[16]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[17]

Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413

[18]

Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124

[19]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079

[20]

Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027

2017 Impact Factor: 0.884

Article outline

Figures and Tables

[Back to Top]