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Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities
Multiple solutions of second-order ordinary differential equation via Morse theory
1. | School of Mathematics Sciences, Shanxi University, Taiyuan, Shanxi 030006, China |
2. | School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083 |
References:
[1] |
R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975. |
[2] |
K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. |
[3] |
K. C. Chang, S. J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problem, Topol. Methods Nonlinear Anal., 3 (1994), 179-187. |
[4] |
Jorge Cossio, Sigifredo Herrón and Carlos Vélez, Existence of solutions for an asymptotically linear Dirichlet problem via Lazer-Solimini results, Topol. Methods Nonlinear Anal., 71 (2009), 66-71.
doi: 10.1016/j.na.2008.10.031. |
[5] |
C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Topol. Methods Nonlinear Anal., 157 (1990), 99-116. |
[6] |
Leszek Gasiński and Nikolaos S. Papageorgiou, A multiplicity theorem for double resonant periodic problems, Advanced Nonlinear Studies, 10 (2010), 819-836. |
[7] |
R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. of the AMS, 311 (1989), 711-726.
doi: 10.1090/s0002-9947-1989-0951886-3. |
[8] |
S. Kesavan, "Nonlinear Functional Analysis," (A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004. |
[9] |
E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance, J. Math. Mech., 19 (1970), 609-623. |
[10] |
A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Topol. Methods Nonlinear Anal., 12 (1988), 761-775. |
[11] |
Wenduan Lu, "Variational Methods in Differential Equations," Sichuan University Publishers, 1995. |
[12] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531.
doi: 10.1007/s00229-007-0127-x. |
[13] |
S. Li and W. Zou, The computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl., 235 (1999), 237-259.
doi: 10.1016/jmaa.1999.6396. |
[14] |
Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.
doi: 10.1016/j.jmaa.2008.12.053. |
[15] |
Shibo Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505.
doi: 10.1016/j.jmaa.2007.01.051. |
[16] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[17] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Issues Math. Ed., vol. 65, 1986. |
[18] |
S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424. |
[19] |
S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance, Electron. J. Differential Equations, 1 (1995), 1-14. |
[20] |
J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.
doi: 10.1016/s0362-54x100100221-2. |
[21] |
J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance, Nonlinear Anal., 70 (2009), 1520-1527.
doi: 10.1016/j.na.2008.04.012. |
[22] |
C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126 (1998), 3263-3270.
doi: 10.1090/S0002-9939-98-04706-6. |
[23] |
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[24] |
Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem, J. Math. Anal. Appl., 279 (2003) 290-307.
doi: 10.1016/S0022-247X(03)00012-X. |
[25] |
W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and morse theory, Journal of Differential Equations, 170 (2001), 68-95.
doi: 10.1006/jdeq.2000.3812. |
show all references
References:
[1] |
R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975. |
[2] |
K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser, Boston, 1993. |
[3] |
K. C. Chang, S. J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problem, Topol. Methods Nonlinear Anal., 3 (1994), 179-187. |
[4] |
Jorge Cossio, Sigifredo Herrón and Carlos Vélez, Existence of solutions for an asymptotically linear Dirichlet problem via Lazer-Solimini results, Topol. Methods Nonlinear Anal., 71 (2009), 66-71.
doi: 10.1016/j.na.2008.10.031. |
[5] |
C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Topol. Methods Nonlinear Anal., 157 (1990), 99-116. |
[6] |
Leszek Gasiński and Nikolaos S. Papageorgiou, A multiplicity theorem for double resonant periodic problems, Advanced Nonlinear Studies, 10 (2010), 819-836. |
[7] |
R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. of the AMS, 311 (1989), 711-726.
doi: 10.1090/s0002-9947-1989-0951886-3. |
[8] |
S. Kesavan, "Nonlinear Functional Analysis," (A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004. |
[9] |
E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance, J. Math. Mech., 19 (1970), 609-623. |
[10] |
A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Topol. Methods Nonlinear Anal., 12 (1988), 761-775. |
[11] |
Wenduan Lu, "Variational Methods in Differential Equations," Sichuan University Publishers, 1995. |
[12] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531.
doi: 10.1007/s00229-007-0127-x. |
[13] |
S. Li and W. Zou, The computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl., 235 (1999), 237-259.
doi: 10.1016/jmaa.1999.6396. |
[14] |
Zhanping Liang and Jiabao Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.
doi: 10.1016/j.jmaa.2008.12.053. |
[15] |
Shibo Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505.
doi: 10.1016/j.jmaa.2007.01.051. |
[16] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[17] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Issues Math. Ed., vol. 65, 1986. |
[18] |
S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424. |
[19] |
S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance, Electron. J. Differential Equations, 1 (1995), 1-14. |
[20] |
J. B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.
doi: 10.1016/s0362-54x100100221-2. |
[21] |
J. B. Su and Leiga Zhao, Multiple periodic solutions of ordinary differential for equations with double resonance, Nonlinear Anal., 70 (2009), 1520-1527.
doi: 10.1016/j.na.2008.04.012. |
[22] |
C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126 (1998), 3263-3270.
doi: 10.1090/S0002-9939-98-04706-6. |
[23] |
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[24] |
Chiara Zanini, Rotation numbers, eigenvalues, and the Poincar-Birkhoff theorem, J. Math. Anal. Appl., 279 (2003) 290-307.
doi: 10.1016/S0022-247X(03)00012-X. |
[25] |
W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and morse theory, Journal of Differential Equations, 170 (2001), 68-95.
doi: 10.1006/jdeq.2000.3812. |
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