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Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain
Infinitely many segregated solutions for coupled nonlinear Schrödinger systems
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China |
$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $ |
$ \mu_1>0 $ |
$ \mu_2>0 $ |
$ \beta\in\mathbb{R} $ |
$ \delta\in\mathbb{R} $ |
$ a(x) $ |
$ b(x) $ |
$ C^\alpha $ |
$ 0<\alpha<1 $ |
$ 0<\delta_0<1 $ |
$ 0<\beta_0<\min\{\mu_1,\mu_2\} $ |
$ 0<\delta<\delta_0 $ |
$ 0<\beta<\beta_0 $ |
References:
[1] |
W. Ao and J. Wei,
Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var., 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[2] |
W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.
doi: 10.1016/j.jfa.2013.06.016. |
[3] |
A. Bahri and Y. Li,
On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$, Revista Mat. Iberoa., 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[4] |
T. Bartsch, N. Dancer and Z. Wang,
A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
T. Bartsch, Z. Wang and J. Wei,
Bound states for a couple Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[6] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[7] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[8] |
B. D. Esry, C. H. Greene, J. P. Burker Jr. and J. L. Bohn,
Hartee-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3584-3597.
|
[9] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry of positive solutions of nonlinear elliptic equations in $ {\Bbb R}^n$, Advances in Math., Supplementary Studies, 7A (1981), 369-402.
|
[10] |
N. Hirano,
Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system, NoDEA Nonlinear Diff. Eq. Appl., 16 (2009), 159-188.
doi: 10.1007/s00030-008-7047-7. |
[11] |
F. Lin, W. Ni and J. Wei,
On the number of interior peak solutions for singularly pertubered Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: 10.1002/cpa.20139. |
[12] |
J. Liu, X. Liu and Z. Wang,
Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var., 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[13] |
Z. Liu and Z. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[14] |
R. Mandel,
Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Diff. Eq. Appl., 22 (2015), 239-262.
doi: 10.1007/s00030-014-0281-2. |
[15] |
S. Peng and Z. Wang,
Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[16] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\Bbb R}^n$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[17] |
N. Soave,
On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var., 53 (2015), 689-718.
doi: 10.1007/s00526-014-0764-3. |
[18] |
Y. Sato and Z. Wang,
On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[19] |
R. Tian and Z. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[20] |
J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied mathematical Science, 189, Springer, London, 2014.
doi: 10.1007/978-1-4471-5526-3. |
[21] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure App. Ana., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
show all references
References:
[1] |
W. Ao and J. Wei,
Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var., 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[2] |
W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.
doi: 10.1016/j.jfa.2013.06.016. |
[3] |
A. Bahri and Y. Li,
On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$, Revista Mat. Iberoa., 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[4] |
T. Bartsch, N. Dancer and Z. Wang,
A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
T. Bartsch, Z. Wang and J. Wei,
Bound states for a couple Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[6] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[7] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[8] |
B. D. Esry, C. H. Greene, J. P. Burker Jr. and J. L. Bohn,
Hartee-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3584-3597.
|
[9] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry of positive solutions of nonlinear elliptic equations in $ {\Bbb R}^n$, Advances in Math., Supplementary Studies, 7A (1981), 369-402.
|
[10] |
N. Hirano,
Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system, NoDEA Nonlinear Diff. Eq. Appl., 16 (2009), 159-188.
doi: 10.1007/s00030-008-7047-7. |
[11] |
F. Lin, W. Ni and J. Wei,
On the number of interior peak solutions for singularly pertubered Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: 10.1002/cpa.20139. |
[12] |
J. Liu, X. Liu and Z. Wang,
Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var., 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[13] |
Z. Liu and Z. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[14] |
R. Mandel,
Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Diff. Eq. Appl., 22 (2015), 239-262.
doi: 10.1007/s00030-014-0281-2. |
[15] |
S. Peng and Z. Wang,
Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[16] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\Bbb R}^n$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[17] |
N. Soave,
On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var., 53 (2015), 689-718.
doi: 10.1007/s00526-014-0764-3. |
[18] |
Y. Sato and Z. Wang,
On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[19] |
R. Tian and Z. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[20] |
J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied mathematical Science, 189, Springer, London, 2014.
doi: 10.1007/978-1-4471-5526-3. |
[21] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure App. Ana., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
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