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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 & 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.  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), 1-8.  Google Scholar

[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.  Google Scholar

[4]

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  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), 409-417.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  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), 243-248. 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), 1278-1287. 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), 1813-1834. 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), 2285-2294. 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), 25-30. 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), 456-463. 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), 1275-1287. 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), 1-8.  Google Scholar

[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.  Google Scholar

[4]

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. 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) 486-490. 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), 85-93. 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), 4883-4892. 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), 409-417.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  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), 243-248. 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), 1278-1287. 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), 1813-1834. 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), 2285-2294. 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), 25-30. 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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

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