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Bifurcation results on positive solutions of an indefinite nonlinear elliptic system
1. | Department of Mathematics & Statistics, Utah State University, Logan, UT 84322, United States |
2. | Department of Mathematics and Statistics, Utah State University, Logan, UT 84322 |
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[3] |
T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branchesof positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, 37, Numbers 3-4 (2010), 345-361. |
[4] |
T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[5] |
T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[6] |
M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[7] |
E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. (3), 30 (1975), 76-94.
doi: 10.1112/plms/s3-30.1.76. |
[8] |
E. N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positivesolutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969. |
[9] |
B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[10] |
H. Kielhöfer, "Bifurcation Theory," Springer-Verlag, New York 2004. |
[11] |
T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equationsin $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[12] |
T. C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439. |
[13] |
Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[14] |
Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193. |
[15] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[16] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[17] |
E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71. |
[18] |
B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS., 138 (2010), 1681-1692.
doi: 10.1090/S0002-9939-10-10231-7. |
[19] |
B. Noris, S. Tavares, H. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302. |
[20] |
S. Oruganti, J. P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting,I: steady states, Transactions of the AMS., 354 (2002), 3601-3619. |
[21] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, Journal of functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[22] |
S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[23] |
S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[24] |
R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223. |
[25] |
J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293. |
[26] |
J. Wei and T. Weth, Radial solutions and phase separation in a system of twocoupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
show all references
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[3] |
T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branchesof positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, 37, Numbers 3-4 (2010), 345-361. |
[4] |
T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[5] |
T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[6] |
M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[7] |
E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. (3), 30 (1975), 76-94.
doi: 10.1112/plms/s3-30.1.76. |
[8] |
E. N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positivesolutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969. |
[9] |
B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[10] |
H. Kielhöfer, "Bifurcation Theory," Springer-Verlag, New York 2004. |
[11] |
T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equationsin $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[12] |
T. C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439. |
[13] |
Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[14] |
Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193. |
[15] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[16] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[17] |
E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71. |
[18] |
B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS., 138 (2010), 1681-1692.
doi: 10.1090/S0002-9939-10-10231-7. |
[19] |
B. Noris, S. Tavares, H. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302. |
[20] |
S. Oruganti, J. P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting,I: steady states, Transactions of the AMS., 354 (2002), 3601-3619. |
[21] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, Journal of functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[22] |
S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[23] |
S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[24] |
R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223. |
[25] |
J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293. |
[26] |
J. Wei and T. Weth, Radial solutions and phase separation in a system of twocoupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
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