American Institute of Mathematical Sciences

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March  2016, 36(3): 1431-1464. doi: 10.3934/dcds.2016.36.1431

Positive solutions of a nonlinear Schrödinger system with nonconstant potentials

Haidong Liu 1, and  Zhaoli Liu 2,

1. 

College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China
2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received  November 2014 Revised  June 2015 Published  August 2015

Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.
Keywords: variational methods., Nonlinear Schrödinger system.
Mathematics Subject Classification: Primary: 35J50; Secondary: 35A1.
Citation: Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
References:
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S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbbR^N$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22. doi: 10.1081/PDE-120002781.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

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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.  Google Scholar

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H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263. doi: 10.1002/cpa.3160230211.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[17]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[22]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[32]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281. doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[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.  Google Scholar

[34]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[35]

R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.  Google Scholar

[36]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbbR^N$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22. doi: 10.1081/PDE-120002781.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrédinger systems, J. Partial Differential Equations, 19 (2006), 200-207.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263. doi: 10.1002/cpa.3160230211.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., No. 596, Springer-Verlag, 1977.  Google Scholar

[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.  Google Scholar

[17]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402.  Google Scholar

[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.  Google Scholar

[19]

F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[32]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281. doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[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.  Google Scholar

[34]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[35]

R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.  Google Scholar

[36]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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