# American Institute of Mathematical Sciences

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

## 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

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 $\mu_j$.
Citation: Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
##### 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF00282048. Google Scholar [12] H. Brezis, On a characterization of flow invariant sets,, Comm. Pure Appl. Math., 23 (1970), 261. 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. 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. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar [15] K. Deimling, Ordinary Differential Equations in Banach Spaces,, Lecture Notes in Math., (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. 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, (1981), 369. Google Scholar [18] F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations,, Phys. Rev. E, 58 (1998), 6700. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0219199708002934. Google Scholar [31] R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar [32] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials,, J. Differential Equations, 227 (2006), 258. 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. 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. 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. Google Scholar [36] E. Timmermans, Phase separation of Bose-Einstein condensates,, Phys. Rev. Lett., 81 (1998), 5718. 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. 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. 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. doi: 10.3934/cpaa.2012.11.1003. Google Scholar [40] M. Willem, Minimax Theorems,, Birkhäuser, (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. doi: 10.1016/j.na.2012.03.027. Google Scholar

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##### 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF00282048. Google Scholar [12] H. Brezis, On a characterization of flow invariant sets,, Comm. Pure Appl. Math., 23 (1970), 261. 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. 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. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar [15] K. Deimling, Ordinary Differential Equations in Banach Spaces,, Lecture Notes in Math., (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. 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, (1981), 369. Google Scholar [18] F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations,, Phys. Rev. E, 58 (1998), 6700. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0219199708002934. Google Scholar [31] R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar [32] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials,, J. Differential Equations, 227 (2006), 258. 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. 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. 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. Google Scholar [36] E. Timmermans, Phase separation of Bose-Einstein condensates,, Phys. Rev. Lett., 81 (1998), 5718. 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. 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. 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. doi: 10.3934/cpaa.2012.11.1003. Google Scholar [40] M. Willem, Minimax Theorems,, Birkhäuser, (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. doi: 10.1016/j.na.2012.03.027. Google Scholar
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