American Institute of Mathematical Sciences

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May  2013, 33(5): 2139-2154. doi: 10.3934/dcds.2013.33.2139

Resonance problems for Kirchhoff type equations

Jijiang Sun 1, and  Chun-Lei Tang 1,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  December 2011 Revised  June 2012 Published  December 2012

The existence of weak solutions is obtained for some Kirchhoff type equations with Dirichlet boundary conditions which are resonant at an arbitrary eigenvalue under a Landesman-Lazer type condition by the minimax methods.
Keywords: Resonance, critical point, Landesman-Lazer type condition, Kirchhoff type problem, link..
Mathematics Subject Classification: Primary: 35J60; Secondary: 47J10,35A1.
Citation: Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
References:
[1]

S. Ahmad, A. C. Lazer and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J., 25 (1976), 933-944.  Google Scholar

[2]

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

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

J. Bouchala and P. Drabek, Strong resonance for some quasilinear elliptic equations, J. Math. Anal. Appl., 245 (2000), 7-19. doi: 10.1006/jmaa.2000.6713.  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]

P. Drábek, On the resonance problem with nonlinearity which has arbitrary linear growth, J. Math Anal. Appl., 127 (1987), 435-442. doi: 10.1016/0022-247X(87)90121-1.  Google Scholar

[7]

P. Drábek and S. B. Robison, Resonance problems for the $p$-Laplacian, J. Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.  Google Scholar

[8]

C. P. Gupta, Solvability of a boundary value problem with the nonlinearity satisfying a sign condition, J. Math. Anal. Appl., 129 (1988), 482-492. doi: 10.1016/0022-247X(88)90266-1.  Google Scholar

[9]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 252 (2011), 1813-1834. doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[10]

E. Landesman and A. Lazer, Nonlinear perturbation of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.  Google Scholar

[11]

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

[12]

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

[13]

J. Mawhin, J. R. Ward and M. Willem, Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem, Proc. Amer. Math. Soc., 93 (1985), 667-674. doi: 10.2307/2045542.  Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.  Google Scholar

[15]

Z. Q. Ou and C. L. Tang, Resonance problems for the $p$-Laplacian systems, J. Math. Anal. Appl., 345 (2008), 511-521. doi: 10.1016/j.jmaa.2008.04.001.  Google Scholar

[16]

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

[17]

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

[18]

E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal., 16 (1991), 455-477. doi: 10.1016/0362-546X(91)90070-H.  Google Scholar

[19]

S. Z. Song and C. L. Tang, Resonance problems for the $p$-Laplacian with a nonlinear boundary condition, Nonlinear Anal., 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

[20]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000.  Google Scholar

[21]

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

[22]

C. L. Tang, Solvability for two-point boundary value problems, J. Math. Anal. Appl., 216 (1997), 368-374. doi: 10.1006/jmaa.1997.5664.  Google Scholar

[23]

C. L. Tang, Solvability of the forced duffing equation at resonance, J. Math. Anal. Appl., 219 (1998), 110-124. doi: 10.1006/jmaa.1997.5793.  Google Scholar

[24]

C. L. Tang, Solvability of Neumann problem for elliptic equations at resonance, Nonlinear Anal., 44 (2001), 323-335. doi: 10.1016/S0362-546X(99)00266-7.  Google Scholar

[25]

J. R. Ward, A boundary value problem with a periodic nonlinearity, Nonlinear Anal., 10 (1986), 207-213. doi: 10.1016/0362-546X(86)90047-7.  Google Scholar

[26]

X. P. Wu and C. L. Tang, Some existence theorems for elliptic resonant problems, J. Math. Anal. Appl., 264 (2001), 133-146. doi: 10.1006/jmaa.2001.7660.  Google Scholar

[27]

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

[28]

X. X. Zhao and C. L. Tang, Resonance problems for $(p,q)$-Laplacian systems, Nonlinear Anal., 72 (2010), 1019-1030. doi: 10.1016/j.na.2009.07.043.  Google Scholar

show all references

References:
[1]

S. Ahmad, A. C. Lazer and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J., 25 (1976), 933-944.  Google Scholar

[2]

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

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

J. Bouchala and P. Drabek, Strong resonance for some quasilinear elliptic equations, J. Math. Anal. Appl., 245 (2000), 7-19. doi: 10.1006/jmaa.2000.6713.  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]

P. Drábek, On the resonance problem with nonlinearity which has arbitrary linear growth, J. Math Anal. Appl., 127 (1987), 435-442. doi: 10.1016/0022-247X(87)90121-1.  Google Scholar

[7]

P. Drábek and S. B. Robison, Resonance problems for the $p$-Laplacian, J. Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.  Google Scholar

[8]

C. P. Gupta, Solvability of a boundary value problem with the nonlinearity satisfying a sign condition, J. Math. Anal. Appl., 129 (1988), 482-492. doi: 10.1016/0022-247X(88)90266-1.  Google Scholar

[9]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 252 (2011), 1813-1834. doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[10]

E. Landesman and A. Lazer, Nonlinear perturbation of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.  Google Scholar

[11]

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

[12]

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

[13]

J. Mawhin, J. R. Ward and M. Willem, Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem, Proc. Amer. Math. Soc., 93 (1985), 667-674. doi: 10.2307/2045542.  Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.  Google Scholar

[15]

Z. Q. Ou and C. L. Tang, Resonance problems for the $p$-Laplacian systems, J. Math. Anal. Appl., 345 (2008), 511-521. doi: 10.1016/j.jmaa.2008.04.001.  Google Scholar

[16]

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

[17]

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

[18]

E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal., 16 (1991), 455-477. doi: 10.1016/0362-546X(91)90070-H.  Google Scholar

[19]

S. Z. Song and C. L. Tang, Resonance problems for the $p$-Laplacian with a nonlinear boundary condition, Nonlinear Anal., 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

[20]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000.  Google Scholar

[21]

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

[22]

C. L. Tang, Solvability for two-point boundary value problems, J. Math. Anal. Appl., 216 (1997), 368-374. doi: 10.1006/jmaa.1997.5664.  Google Scholar

[23]

C. L. Tang, Solvability of the forced duffing equation at resonance, J. Math. Anal. Appl., 219 (1998), 110-124. doi: 10.1006/jmaa.1997.5793.  Google Scholar

[24]

C. L. Tang, Solvability of Neumann problem for elliptic equations at resonance, Nonlinear Anal., 44 (2001), 323-335. doi: 10.1016/S0362-546X(99)00266-7.  Google Scholar

[25]

J. R. Ward, A boundary value problem with a periodic nonlinearity, Nonlinear Anal., 10 (1986), 207-213. doi: 10.1016/0362-546X(86)90047-7.  Google Scholar

[26]

X. P. Wu and C. L. Tang, Some existence theorems for elliptic resonant problems, J. Math. Anal. Appl., 264 (2001), 133-146. doi: 10.1006/jmaa.2001.7660.  Google Scholar

[27]

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

[28]

X. X. Zhao and C. L. Tang, Resonance problems for $(p,q)$-Laplacian systems, Nonlinear Anal., 72 (2010), 1019-1030. doi: 10.1016/j.na.2009.07.043.  Google Scholar

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