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Variational methods for non-local operators of elliptic type
Resonance problems for Kirchhoff type equations
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
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. |
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