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Lyapunov inequalities for partial differential equations at radial higher eigenvalues
Existence and qualitative properties of solutions for nonlinear Dirichlet problems
1. | Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States |
2. | Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellĺn, Apartado Aéreo 3840, Medellín, Colombia |
3. | Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Apartado Aéreo 3840, Medellín, Colombia |
References:
[1] |
R. A. Adams, "Sobolev Spaces," New York, AcademicPress, 1975. |
[2] |
A. Aftalion and F. Pacella, Qualitative propertiesof nodal solutions of semilinear elliptic equations in radiallysymmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344.
doi: 10.1016/j.crma.2004.07.004. |
[3] |
T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations, Comm. Partial Diff. Eq., 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch, K. C. Chang and Z. Q. Wang, On the Morse indices of sign changing solutions of nonlinear ellipticproblems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[5] |
T. Bartsch and Z. Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131. |
[6] |
T. Bartsch and T. Weth, A note on additional properties ofsign-changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. |
[7] |
H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems, Comm. in Partial Diff. Equations, 2 (1977), 601-614. |
[8] |
A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561.
doi: 10.1137/S0036141092230106. |
[9] |
A. Castro, J. Cossio, and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. M., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[10] |
A. Castro, J. Cossio and J. M. Neuberger, A minmax principle,index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electronic Journal Diff. Eqns., 1998 (1998), 1-18. |
[11] |
A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem, Annali di Mat. Pura et Appl., to appear. |
[12] |
A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem, 101-107 (electronic), Electron. J. Diff. Eqns., Conf. 10, Southwest Texas State Univ., San Marcos, TX, 2003. |
[13] |
K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems," Birkhäuser, Boston, 1993. |
[14] |
J. Cossio and S. Herrón, Nontrivial Solutions for asemilinear Dirichlet problem with nonlinearity crossing multipleeigenvalues, Journal of Dynamics and Differential Equations, 16 (2004), 795-803.
doi: 10.1007/s10884-004-6695-5. |
[15] |
J. Cossio, S. Herrón and C. Vélez, Existence of solutions for an asymptotically linearDirichlet problem via Lazer-Solimini results, Nonlinear Anal., 71 (2009), 66-71.
doi: 10.1016/j.na.2008.10.031. |
[16] |
A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 70 (1979), 113-137. |
[17] |
K. Chang, S. Li and J. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. Methods in Nonlinear Anal., 3 (1994), 179-187. |
[18] |
J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal, Rev. Colombiana Mat., 37 (2003), 25-36. |
[19] |
E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Mathematische Annalen, 272 (1985), 421-440.
doi: 10.1007/BF01455568. |
[20] |
D. De Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. |
[21] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 0.1007/BF01221125. |
[22] |
V. Guillemin and A. Pollack, "Differential Topology," New York, NY, Prentice-Hall, 1974. |
[23] |
D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin 1977. |
[24] |
H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. |
[25] |
S. Kesavan, "Nonlinear Functional Analysis (A First Course)," Text and readings in mathematics 28. Hindustan Book Agency, India, 2004. |
[26] |
A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal., 12 (1988), 761-775.
doi: 10.1016/0362-546X(88)90037-5. |
[27] |
S. 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. |
[28] |
Z. Liu, F. A. van Heerden, and Z. Q. Wang, Nodal type bound states of Schrödinger equations via invariant sets and minimax methods, Journal of Differential Equations, 214 (2005), 358-390.
doi: 10.1016/j.jde.2004.08.023. |
[29] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," Regional Conference Series in Mathematics, number 65, AMS, Providence, R.I., 1986. |
[30] |
X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, Journal of Functional Analysis, 160 (1998), 223-253.
doi: 10.1006/jfan.1998.3301. |
show all references
References:
[1] |
R. A. Adams, "Sobolev Spaces," New York, AcademicPress, 1975. |
[2] |
A. Aftalion and F. Pacella, Qualitative propertiesof nodal solutions of semilinear elliptic equations in radiallysymmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344.
doi: 10.1016/j.crma.2004.07.004. |
[3] |
T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations, Comm. Partial Diff. Eq., 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch, K. C. Chang and Z. Q. Wang, On the Morse indices of sign changing solutions of nonlinear ellipticproblems, Math. Z., 233 (2000), 655-677.
doi: 10.1007/s002090050492. |
[5] |
T. Bartsch and Z. Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131. |
[6] |
T. Bartsch and T. Weth, A note on additional properties ofsign-changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. |
[7] |
H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems, Comm. in Partial Diff. Equations, 2 (1977), 601-614. |
[8] |
A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561.
doi: 10.1137/S0036141092230106. |
[9] |
A. Castro, J. Cossio, and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. M., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[10] |
A. Castro, J. Cossio and J. M. Neuberger, A minmax principle,index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electronic Journal Diff. Eqns., 1998 (1998), 1-18. |
[11] |
A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem, Annali di Mat. Pura et Appl., to appear. |
[12] |
A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem, 101-107 (electronic), Electron. J. Diff. Eqns., Conf. 10, Southwest Texas State Univ., San Marcos, TX, 2003. |
[13] |
K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems," Birkhäuser, Boston, 1993. |
[14] |
J. Cossio and S. Herrón, Nontrivial Solutions for asemilinear Dirichlet problem with nonlinearity crossing multipleeigenvalues, Journal of Dynamics and Differential Equations, 16 (2004), 795-803.
doi: 10.1007/s10884-004-6695-5. |
[15] |
J. Cossio, S. Herrón and C. Vélez, Existence of solutions for an asymptotically linearDirichlet problem via Lazer-Solimini results, Nonlinear Anal., 71 (2009), 66-71.
doi: 10.1016/j.na.2008.10.031. |
[16] |
A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 70 (1979), 113-137. |
[17] |
K. Chang, S. Li and J. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. Methods in Nonlinear Anal., 3 (1994), 179-187. |
[18] |
J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal, Rev. Colombiana Mat., 37 (2003), 25-36. |
[19] |
E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Mathematische Annalen, 272 (1985), 421-440.
doi: 10.1007/BF01455568. |
[20] |
D. De Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. |
[21] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 0.1007/BF01221125. |
[22] |
V. Guillemin and A. Pollack, "Differential Topology," New York, NY, Prentice-Hall, 1974. |
[23] |
D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin 1977. |
[24] |
H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. |
[25] |
S. Kesavan, "Nonlinear Functional Analysis (A First Course)," Text and readings in mathematics 28. Hindustan Book Agency, India, 2004. |
[26] |
A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal., 12 (1988), 761-775.
doi: 10.1016/0362-546X(88)90037-5. |
[27] |
S. 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. |
[28] |
Z. Liu, F. A. van Heerden, and Z. Q. Wang, Nodal type bound states of Schrödinger equations via invariant sets and minimax methods, Journal of Differential Equations, 214 (2005), 358-390.
doi: 10.1016/j.jde.2004.08.023. |
[29] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," Regional Conference Series in Mathematics, number 65, AMS, Providence, R.I., 1986. |
[30] |
X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, Journal of Functional Analysis, 160 (1998), 223-253.
doi: 10.1006/jfan.1998.3301. |
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