2013, 12(6): 2773-2786. doi: 10.3934/cpaa.2013.12.2773

Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China

Received  October 2012 Revised  February 2013 Published  May 2013

In the present paper, the existence and multiplicity of solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value conditions are obtained via the variational method.
Citation: 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
References:
[1]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011.

[2]

A. Hamydy, M. Massar and N. Tsouli, Existence of solution for p-Kirchhoff type problems with critical exponents,, Electronic J. Differential Equations, 105 (2011), 1.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 41 (1973), 349. doi: 10.1016/0022-1236(73)90051-7.

[4]

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, J. Math. Anal. Appl., 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025.

[5]

H. Brezis and E. Lieb, A relation between pointwise conergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3.

[6]

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

[7]

C. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065.

[8]

C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differential Equation and Applications, 23 (2010), 409.

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[10]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $R^3$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059.

[11]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, Nonlinear Anal., 75 (2012), 3470. doi: 10.1016/j.na.2012.01.004.

[12]

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. doi: 10.1016/j.jde.2012.05.023.

[13]

J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations,, Nonlinear Anal., 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061.

[14]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang-index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006,.

[15]

L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-120-0536-1.

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Application to Differetial Equations,", in: CBMS Reg. Conf. Series. Math. 65, (1986).

[17]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées,, Bull. Acad. Sci. URSS, 17-26 (1940), 17.

[18]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS) \textbf{138} (1975), 138 (1975), 152. doi: 10.1070/SM1975v025n01ABEH002203.

[19]

T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1,.

[20]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^3$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023.

[21]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $R^3$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035.

[22]

Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, J. Differential Equations, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017.

[23]

Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems,, Nonlinear Anal., 73 (2010), 25. doi: 10.1016/j.na.2010.02.008.

[24]

Z. T. Zhang and K. Perera, Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102.

show all references

References:
[1]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011.

[2]

A. Hamydy, M. Massar and N. Tsouli, Existence of solution for p-Kirchhoff type problems with critical exponents,, Electronic J. Differential Equations, 105 (2011), 1.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 41 (1973), 349. doi: 10.1016/0022-1236(73)90051-7.

[4]

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, J. Math. Anal. Appl., 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025.

[5]

H. Brezis and E. Lieb, A relation between pointwise conergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3.

[6]

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

[7]

C. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065.

[8]

C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differential Equation and Applications, 23 (2010), 409.

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[10]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $R^3$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059.

[11]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, Nonlinear Anal., 75 (2012), 3470. doi: 10.1016/j.na.2012.01.004.

[12]

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. doi: 10.1016/j.jde.2012.05.023.

[13]

J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations,, Nonlinear Anal., 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061.

[14]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang-index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006,.

[15]

L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-120-0536-1.

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Application to Differetial Equations,", in: CBMS Reg. Conf. Series. Math. 65, (1986).

[17]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées,, Bull. Acad. Sci. URSS, 17-26 (1940), 17.

[18]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS) \textbf{138} (1975), 138 (1975), 152. doi: 10.1070/SM1975v025n01ABEH002203.

[19]

T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1,.

[20]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^3$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023.

[21]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $R^3$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035.

[22]

Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, J. Differential Equations, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017.

[23]

Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems,, Nonlinear Anal., 73 (2010), 25. doi: 10.1016/j.na.2010.02.008.

[24]

Z. T. Zhang and K. Perera, Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

V. V. Motreanu. Uniqueness results for a Dirichlet problem with variable exponent. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1399-1410. doi: 10.3934/cpaa.2010.9.1399

[6]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111

[7]

Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009

[8]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845

[9]

Isabel Flores. Singular solutions of the Brezis-Nirenberg problem in a ball. Communications on Pure & Applied Analysis, 2009, 8 (2) : 673-682. doi: 10.3934/cpaa.2009.8.673

[10]

Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951

[11]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567

[12]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[13]

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

[14]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095

[15]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701

[16]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[17]

Jan Andres, Luisa Malaguti, Martina Pavlačková. Hartman-type conditions for multivalued Dirichlet problem in abstract spaces. Conference Publications, 2015, 2015 (special) : 38-55. doi: 10.3934/proc.2015.0038

[18]

Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645

[19]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233

[20]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]