# American Institute of Mathematical Sciences

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

## Resonance problems for Kirchhoff type equations

 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.
Citation: Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and 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. [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. [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. [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. [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] 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. [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. [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. [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. [10] E. Landesman and A. Lazer, Nonlinear perturbation of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623. [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. [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. [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. [14] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. [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. [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. [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. [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. [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. [20] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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] 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. [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. [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. [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. [10] E. Landesman and A. Lazer, Nonlinear perturbation of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623. [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. [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. [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. [14] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," in: Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. [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. [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. [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. [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. [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. [20] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000. [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. [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. [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. [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. [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. [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. [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. [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.
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