November  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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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,.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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,.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[2]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[3]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[4]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[5]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[6]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[7]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[8]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392

[9]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292

[10]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287

[11]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[12]

Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021004

[13]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[14]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[16]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[17]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[18]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[19]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

[20]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (32)

Other articles
by authors

[Back to Top]