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Existence of piecewise linear Lyapunov functions in arbitrary dimensions
Perturbed elliptic equations with oscillatory nonlinearities
1. | School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China |
2. | School of Mathematical Sciences, Capital Normal University, Beijing 100048, China |
References:
[1] |
G. Anello and G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations, 234 (2007), 80-90.
doi: 10.1016/j.jde.2006.11.011. |
[2] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[3] |
T. Bartsch and Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differential Equations, 9 (2004), 645-676. |
[4] |
T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the $p$-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258. |
[5] |
T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[6] |
T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc. (3), 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[7] |
S. Chen and S. Li, Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity, Calc. Var. Partial Differential Equations, 27 (2006), 105-123.
doi: 10.1007/s00526-006-0025-1. |
[8] |
L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and montonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[9] |
P. Habets, E. Serra and M. Tarallo, Multiplicity results for boundary value problems with potentials oscillating around resonance, J. Differential Equations, 138 (1997), 133-156.
doi: 10.1006/jdeq.1997.3267. |
[10] |
A. Kristály, Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Differential Equations, 245 (2008), 3849-3868.
doi: 10.1016/j.jde.2008.05.014. |
[11] |
S. Li and Z. Liu, Perturbations from symmeric elliptic boundary value problems, J. Differential Equations, 185 (2002), 271-280.
doi: 10.1006/jdeq.2001.4160. |
[12] |
Y. Li, Z. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68. |
[13] |
P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733.
doi: 10.1080/03605309608821205. |
[14] |
J. Saint Raymond, On the multiplicity of the solutions of equations $-\Delta u=\lambda f(u)$, J. Differential Equations, 180 (2002), 65-88.
doi: 10.1006/jdeq.2001.4057. |
[15] |
R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859.
doi: 10.1090/S0002-9947-1988-0933322-5. |
[16] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
show all references
References:
[1] |
G. Anello and G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations, 234 (2007), 80-90.
doi: 10.1016/j.jde.2006.11.011. |
[2] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[3] |
T. Bartsch and Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differential Equations, 9 (2004), 645-676. |
[4] |
T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the $p$-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258. |
[5] |
T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[6] |
T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc. (3), 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[7] |
S. Chen and S. Li, Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity, Calc. Var. Partial Differential Equations, 27 (2006), 105-123.
doi: 10.1007/s00526-006-0025-1. |
[8] |
L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and montonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[9] |
P. Habets, E. Serra and M. Tarallo, Multiplicity results for boundary value problems with potentials oscillating around resonance, J. Differential Equations, 138 (1997), 133-156.
doi: 10.1006/jdeq.1997.3267. |
[10] |
A. Kristály, Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Differential Equations, 245 (2008), 3849-3868.
doi: 10.1016/j.jde.2008.05.014. |
[11] |
S. Li and Z. Liu, Perturbations from symmeric elliptic boundary value problems, J. Differential Equations, 185 (2002), 271-280.
doi: 10.1006/jdeq.2001.4160. |
[12] |
Y. Li, Z. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68. |
[13] |
P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733.
doi: 10.1080/03605309608821205. |
[14] |
J. Saint Raymond, On the multiplicity of the solutions of equations $-\Delta u=\lambda f(u)$, J. Differential Equations, 180 (2002), 65-88.
doi: 10.1006/jdeq.2001.4057. |
[15] |
R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859.
doi: 10.1090/S0002-9947-1988-0933322-5. |
[16] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
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