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March  2014, 13(2): 483-494. doi: 10.3934/cpaa.2014.13.483

Nontrivial solutions for Kirchhoff type equations via Morse theory

Jijiang Sun 1, and  Shiwang Ma 1,

1. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received  March 2012 Revised  August 2013 Published  October 2013

In this paper, the existence of nontrivial solutions is obtained for a class of Kirchhoff type problems with Dirichlet boundary conditions by computing the critical groups and Morse theory.
Keywords: critical groups, Morse theory, local linking., Kirchhoff type problems.
Mathematics Subject Classification: Primary: 35J60; Secondary: 58E05, 47J30, 35J2.
Citation: Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure & Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483
References:
[1]

C. O. Alves, F. J. S. A. Correa 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

[2]

T. Bartsch and S. J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[3]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian) Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.  Google Scholar

[4]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[5]

C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[6]

F. Fang and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[7]

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 (2011), 1813-1834. doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[8]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[9]

Q. S. Jiu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal.Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar

[10]

G. B. Li and C. Y. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037.  Google Scholar

[11]

C. G. Liu and Y. Q. Zheng, Linking solutions for $p$-Laplace equations with nonlinear boundary conditions and indefinite weight, Calc. Var. Partial Differential Equations, 41 (2011), 261-284. doi: 10.1007/s00526-010-0361-z.  Google Scholar

[12]

J. Q. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.  Google Scholar

[13]

J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[14]

S. B. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.  Google Scholar

[15]

S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Mathematica Sinica, 46 (2003), 625-630 (Chinese).  Google Scholar

[16]

T. F. Ma and J. E. Muñoz 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

[17]

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

[18]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[19]

K. Perera, R. P. Agarwal and D. O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Amer. Math. Soc, Providence, RI, 2010.  Google Scholar

[20]

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

[21]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[22]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.  Google Scholar

[23]

J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154. doi: 10.3934/dcds.2013.33.2139.  Google Scholar

[24]

J. Wang and C. L. Tang, Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations, Boundary Value Probl., 2006 (2006), 1-12. doi: 10.1155/BVP/2006/47275.  Google Scholar

[25]

Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire., 8 (1991), 43-57.  Google Scholar

[26]

M. Willem, "Minimax Theorems," Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[27]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.  Google Scholar

[28]

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]

C. O. Alves, F. J. S. A. Correa 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

[2]

T. Bartsch and S. J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[3]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian) Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.  Google Scholar

[4]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[5]

C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[6]

F. Fang and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[7]

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 (2011), 1813-1834. doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[8]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[9]

Q. S. Jiu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal.Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar

[10]

G. B. Li and C. Y. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037.  Google Scholar

[11]

C. G. Liu and Y. Q. Zheng, Linking solutions for $p$-Laplace equations with nonlinear boundary conditions and indefinite weight, Calc. Var. Partial Differential Equations, 41 (2011), 261-284. doi: 10.1007/s00526-010-0361-z.  Google Scholar

[12]

J. Q. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.  Google Scholar

[13]

J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[14]

S. B. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.  Google Scholar

[15]

S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Mathematica Sinica, 46 (2003), 625-630 (Chinese).  Google Scholar

[16]

T. F. Ma and J. E. Muñoz 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

[17]

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

[18]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[19]

K. Perera, R. P. Agarwal and D. O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Amer. Math. Soc, Providence, RI, 2010.  Google Scholar

[20]

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

[21]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[22]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.  Google Scholar

[23]

J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154. doi: 10.3934/dcds.2013.33.2139.  Google Scholar

[24]

J. Wang and C. L. Tang, Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations, Boundary Value Probl., 2006 (2006), 1-12. doi: 10.1155/BVP/2006/47275.  Google Scholar

[25]

Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire., 8 (1991), 43-57.  Google Scholar

[26]

M. Willem, "Minimax Theorems," Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[27]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.  Google Scholar

[28]

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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