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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

Qi-Lin Xie 1, , Xing-Ping Wu 1, and  Chun-Lei Tang 1,

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.
Keywords: Dirichlet problem, Kirchhoff type problem, Brezis-Lieb Lemma., critical exponent.
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 and 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), 1275-1287. 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), 1-8.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381. 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), 488-495. 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) 486-490. 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), 85-93. 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), 4883-4892. 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), 409-417.

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) 437-477. 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). 564-574. 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), 3470-3479. 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), 2314-2351. 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), 1212-1222. 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), 246-255. 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), 473-487. 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, Amer. Math. Soc., Providence, RI, 1986.

[17]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Sér 17-26 (1940), (Izvestia Akad. Nauk SSSR) 313-345.

[18]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS) 138 (1975), 152-166, 168 (in Russian). 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), 243-248. 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), 1278-1287. 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), 1813-1834. 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), 2285-2294. 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), 25-30. 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), 456-463. 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), 1275-1287. 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), 1-8.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381. 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), 488-495. 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) 486-490. 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), 85-93. 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), 4883-4892. 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), 409-417.

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) 437-477. 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). 564-574. 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), 3470-3479. 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), 2314-2351. 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), 1212-1222. 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), 246-255. 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), 473-487. 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, Amer. Math. Soc., Providence, RI, 1986.

[17]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Sér 17-26 (1940), (Izvestia Akad. Nauk SSSR) 313-345.

[18]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS) 138 (1975), 152-166, 168 (in Russian). 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), 243-248. 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), 1278-1287. 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), 1813-1834. 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), 2285-2294. 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), 25-30. 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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

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