American Institute of Mathematical Sciences

Advanced
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 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.

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.

[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.

[1]

Maria Do Rosario Grossinho, Rogério Martins. Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions. Conference Publications, 2001, 2001 (Special) : 174-181. doi: 10.3934/proc.2001.2001.174
[2]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[3]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[4]

Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027
[5]

Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078
[6]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080
[7]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111
[8]

Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009
[9]

Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124
[10]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
[11]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
[12]

Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
[13]

Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure and Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008
[14]

Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074
[15]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
[16]

Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 757-767. doi: 10.3934/dcds.2011.29.757
[17]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[18]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure and Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
[19]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5465-5494. doi: 10.3934/dcdsb.2020354
[20]

Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1351-1370. doi: 10.3934/cpaa.2016.15.1351

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy