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Positive solutions of a nonlinear Schrödinger system with nonconstant potentials
1. | College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China |
2. | School of Mathematical Sciences, Capital Normal University, Beijing 100048, China |
References:
[1] |
S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbb{R}^N2$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22.
doi: 10.1081/PDE-120002781. |
[2] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[3] |
A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb R^n$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[4] |
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. |
[5] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[6] |
A. Bahri and Y. Y. Li, On the min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb R^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[7] |
A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413.
doi: 10.1016/S0294-1449(97)80142-4. |
[8] |
T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[9] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrédinger systems, J. Partial Differential Equations, 19 (2006), 200-207. |
[10] |
T. Bartsch, Z.-Q. Wang and J.C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[11] |
V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domain, Arch. Rational Mech. Anal., 99 (1987), 283-300.
doi: 10.1007/BF00282048. |
[12] |
H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263.
doi: 10.1002/cpa.3160230211. |
[13] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[14] |
E. N. Dancer, J. C. 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. |
[15] |
K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., No. 596, Springer-Verlag, 1977. |
[16] |
W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equations, Arch. Rational Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[17] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402. |
[18] |
F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707.
doi: 10.1103/PhysRevE.58.6700. |
[19] |
F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155. |
[20] |
F. T. Hioe and T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
doi: 10.1088/0305-4470/35/42/303. |
[21] |
J. Hirata, A positive solution of a nonlinear Schrödinger equation with $G$-symmetry, Nonlinear Anal., 69 (2008), 3174-3189.
doi: 10.1016/j.na.2007.09.010. |
[22] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[23] |
L. S. Lin, Z. L. Liu and S. W. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689.
doi: 10.1512/iumj.2009.58.3611. |
[24] |
T.-C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n,\ n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[25] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[26] |
H. D. Liu and Z. L. Liu, Ground states of a nonlinear Schrödinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257-278.
doi: 10.1007/s11425-014-4914-z. |
[27] |
Z. L. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[28] |
Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193. |
[29] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[30] |
L. A. Maia, E. Montefusco and B. Pellacci, Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Comm. Contemp. Math., 10 (2008), 651-669.
doi: 10.1142/S0219199708002934. |
[31] |
R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[32] |
A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[33] |
Y. Sato and Z.-Q. 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. |
[34] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[35] |
R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. |
[36] |
E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
doi: 10.1103/PhysRevLett.81.5718. |
[37] |
J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[38] |
J. C. Wei and T. Weth, Radial solutions and phase seperation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[39] |
J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[40] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[41] |
T. F. Wu, Two coupled nonlinear Schrödinger equations involving a nonconstant coupling coefficient, Nonlinear Anal., 75 (2012), 4766-4783.
doi: 10.1016/j.na.2012.03.027. |
show all references
References:
[1] |
S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbb{R}^N2$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22.
doi: 10.1081/PDE-120002781. |
[2] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
doi: 10.1103/PhysRevLett.82.2661. |
[3] |
A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb R^n$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[4] |
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. |
[5] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[6] |
A. Bahri and Y. Y. Li, On the min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb R^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[7] |
A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413.
doi: 10.1016/S0294-1449(97)80142-4. |
[8] |
T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[9] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrédinger systems, J. Partial Differential Equations, 19 (2006), 200-207. |
[10] |
T. Bartsch, Z.-Q. Wang and J.C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[11] |
V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domain, Arch. Rational Mech. Anal., 99 (1987), 283-300.
doi: 10.1007/BF00282048. |
[12] |
H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263.
doi: 10.1002/cpa.3160230211. |
[13] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[14] |
E. N. Dancer, J. C. 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. |
[15] |
K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., No. 596, Springer-Verlag, 1977. |
[16] |
W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equations, Arch. Rational Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[17] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402. |
[18] |
F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707.
doi: 10.1103/PhysRevE.58.6700. |
[19] |
F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155. |
[20] |
F. T. Hioe and T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
doi: 10.1088/0305-4470/35/42/303. |
[21] |
J. Hirata, A positive solution of a nonlinear Schrödinger equation with $G$-symmetry, Nonlinear Anal., 69 (2008), 3174-3189.
doi: 10.1016/j.na.2007.09.010. |
[22] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[23] |
L. S. Lin, Z. L. Liu and S. W. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689.
doi: 10.1512/iumj.2009.58.3611. |
[24] |
T.-C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n,\ n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[25] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[26] |
H. D. Liu and Z. L. Liu, Ground states of a nonlinear Schrödinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257-278.
doi: 10.1007/s11425-014-4914-z. |
[27] |
Z. L. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[28] |
Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193. |
[29] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[30] |
L. A. Maia, E. Montefusco and B. Pellacci, Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Comm. Contemp. Math., 10 (2008), 651-669.
doi: 10.1142/S0219199708002934. |
[31] |
R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[32] |
A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[33] |
Y. Sato and Z.-Q. 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. |
[34] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[35] |
R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. |
[36] |
E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
doi: 10.1103/PhysRevLett.81.5718. |
[37] |
J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[38] |
J. C. Wei and T. Weth, Radial solutions and phase seperation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[39] |
J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[40] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[41] |
T. F. Wu, Two coupled nonlinear Schrödinger equations involving a nonconstant coupling coefficient, Nonlinear Anal., 75 (2012), 4766-4783.
doi: 10.1016/j.na.2012.03.027. |
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