Resonance problems for Kirchhoff type equations

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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

Abstract
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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 - A, 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.2307/2045542. Google Scholar
[14]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", in: Applied Mathematical Sciences, 74 (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. 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. 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 (1986). Google Scholar
[18]	E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance,, Nonlinear Anal., 16 (1991), 455. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.na.2009.07.043. Google Scholar

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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.2307/2045542. Google Scholar
[14]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", in: Applied Mathematical Sciences, 74 (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. 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. 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 (1986). Google Scholar
[18]	E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance,, Nonlinear Anal., 16 (1991), 455. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.na.2009.07.043. Google Scholar

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