Nontrivial solutions for Kirchhoff type equations via Morse theory

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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

Abstract
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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 and 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.
[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.
[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.
[4]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Q. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.
[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.
[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.
[15]	S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Mathematica Sinica, 46 (2003), 625-630 (Chinese).
[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.
[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.
[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.
[19]	K. Perera, R. P. Agarwal and D. O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Amer. Math. Soc, Providence, RI, 2010.
[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.
[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.
[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.
[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.
[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.
[25]	Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire., 8 (1991), 43-57.
[26]	M. Willem, "Minimax Theorems," Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[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.
[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.

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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.
[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.
[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.
[4]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Q. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39.
[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.
[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.
[15]	S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Mathematica Sinica, 46 (2003), 625-630 (Chinese).
[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.
[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.
[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.
[19]	K. Perera, R. P. Agarwal and D. O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Amer. Math. Soc, Providence, RI, 2010.
[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.
[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.
[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.
[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.
[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.
[25]	Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire., 8 (1991), 43-57.
[26]	M. Willem, "Minimax Theorems," Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[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.
[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.

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