# American Institute of Mathematical Sciences

February  2020, 19(2): 1181-1204. doi: 10.3934/cpaa.2020055

## Multiple solutions for a nonlinear Schrödinger systems

 Department of Mathematics, Tsinghua University, Beijing, 100084, China

* Corresponding author

Received  May 2019 Revised  May 2019 Published  October 2019

Fund Project: This work is supported by NSFC (11571040, 11771235).

This paper is concerned with the following nonlinear Schrödinger systems:
 \left\{\begin{aligned}-\Delta u+a(x)u& = |u|^{p-2}u+\beta |u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}} \ \ \hbox{in } \mathbb{R}^{N}\\-\Delta v+a(x)v& = |v|^{p-2}v+\beta |v|^{\frac{p}{2}-2}v|u|^{\frac{p}{2}}\ \ \ \hbox{in } \mathbb{R}^{N}\\(u,v)&\in (H^1(\mathbb{R}^N))^2,\end{aligned}\right. \ \ \ \ \ \ \ {(P)}
where
 $N\geq3$
and
 $2 , $ \beta\in \mathbb{R} $is a coupling constant. $ a(x) $is a $ \mathcal{C}^1 $potential function. In the repulsive case, i.e. $ \beta<0 $, under some suitable decay assumptions but without any symmetric assumptions on the potential $ a(x) $, we prove the existence of infinitely many solutions for the problem $ (P). $Citation: Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055 ##### References:  [1] R. Adams, Sobolev Spaces, Academic press, New York-London, 1975. [2] W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equations, 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5. [3] A. Bahri and P. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Ins. H. Poincaré Anayse Nonlinéaire, 14 (1997), 285-300. doi: 10.1016/S0294-1449(97)80142-4. [4] T. Bartsch and Z. Wang, Sign changing solutions of nonlinear Schrödinger equations, Top. Meth. Nonlinear Anal., 13 (1999), 191-198. doi: 10.12775/TMNA.1999.010. [5] T. Bartsch, Z. Wang and J. Wei, Bound states for a coupled Schr$\ddot{o}$dinger system, J. Fixed Point Theory Appl., 2 (2007), 67-82. doi: 10.1007/s11784-007-0033-6. [6] T. Bartsch and M. Willelm, Infinitely many nonradial solutions of an Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460. doi: 10.1006/jfan.1993.1133. [7] V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048. [8] H. Berestycki and P. Lions, Nonlinear scalar field equations, Ⅰ Existence of a ground state. Ⅱ Existence of infinitely many solutions, Arch. Rat. Mech. Anal., 82 (1983), 313-346,347-376. doi: 10.1007/BF00250556. [9] G. Cerami, D. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar fileld equation, Calc. Var. Partial Differential Equations, 23 (2005), 139-168. doi: 10.1007/s00526-004-0293-6. [10] G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar fileld equation, Comm. Pure Appl. math., 66 (2013), 372-413. doi: 10.1002/cpa.21410. [11] M. Conti, S. Terracini and G. Verzini, Neharis problem and competing species systems, Ann. Inst. H. Poincar Ana Non Linnaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. [12] 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. Poincar Ana. Non Linaire, 27 (2010), 953-869. doi: 10.1016/j.anihpc.2010.01.009. [13] G. Devillanova and S. Solimini, Concentrations estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 7 (2002), 1257-1280. [14] W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336. [15] B. Esry, C. Greene, J. Burke Jr. and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. [16] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1983. doi: 10.1007/978-3-642-61798-0. [17] Q. Han and F. Lin, Elliptic Paritial Differential Equations,$2^nd$edition, AMS, 2011. [18] P. L. Lions, The concentration compactness principle in the calculus of variations Parts Ⅰ and Ⅱ, Ann. Ins. H. Poincaré, 1 (1984), 109-145, 223-283. [19] T. Lin and J. Wei, Ground state of$N$couple nonlinear Schrödinger equations in$\mathbb{R}^n, n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. [20] T. C. Lin and J. Wei, Spike in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincar Ana. Non Linaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [21] C. Maniscalco, Multiple solutions for semilinear elliptic problems in$\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110. [22] Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235. [23] E. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform holder bounds for nonlinear Schrödinger system with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309. [24] R. Palais, Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132. doi: 10.1016/0040-9383(66)90013-9. [25] S. Peng and Z. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger system, Arch. Rat. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0. [26] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [27] Y. Rudyak and F. Schlenk, Lusternik-Schnirelmann theory for fixed points of maps, Topological Methods in Nonlinear Analysis, 21 (2003), 171-194. doi: 10.12775/TMNA.2003.011. [28] S. Solimini, Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453. doi: 10.1007/BF01171757. [29] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. [30] S. Terracini and G. Verzini, Multiple phase in$k-$mixture of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y. [31] J. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equation, Rend. Lincei Mat. Appl., 18 (2007), 279-293. doi: 10.4171/RLM/495. [32] J. Wei and T. Weth, Radial solutions and phase sepration in a system of two coupled systems of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9. [33] J. 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. [34] C. Zelati and P. Robinowitz, Homoclinic type solutions for a semilinear elliptic PDE on$\mathbb{R}^N$, Comm. Pure. Appl. Math., 10 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. show all references ##### References:  [1] R. Adams, Sobolev Spaces, Academic press, New York-London, 1975. [2] W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equations, 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5. [3] A. Bahri and P. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Ins. H. Poincaré Anayse Nonlinéaire, 14 (1997), 285-300. doi: 10.1016/S0294-1449(97)80142-4. [4] T. Bartsch and Z. Wang, Sign changing solutions of nonlinear Schrödinger equations, Top. Meth. Nonlinear Anal., 13 (1999), 191-198. doi: 10.12775/TMNA.1999.010. [5] T. Bartsch, Z. Wang and J. Wei, Bound states for a coupled Schr$\ddot{o}$dinger system, J. Fixed Point Theory Appl., 2 (2007), 67-82. doi: 10.1007/s11784-007-0033-6. [6] T. Bartsch and M. Willelm, Infinitely many nonradial solutions of an Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460. doi: 10.1006/jfan.1993.1133. [7] V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048. [8] H. Berestycki and P. Lions, Nonlinear scalar field equations, Ⅰ Existence of a ground state. Ⅱ Existence of infinitely many solutions, Arch. Rat. Mech. Anal., 82 (1983), 313-346,347-376. doi: 10.1007/BF00250556. [9] G. Cerami, D. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar fileld equation, Calc. Var. Partial Differential Equations, 23 (2005), 139-168. doi: 10.1007/s00526-004-0293-6. [10] G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar fileld equation, Comm. Pure Appl. math., 66 (2013), 372-413. doi: 10.1002/cpa.21410. [11] M. Conti, S. Terracini and G. Verzini, Neharis problem and competing species systems, Ann. Inst. H. Poincar Ana Non Linnaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. [12] 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. Poincar Ana. Non Linaire, 27 (2010), 953-869. doi: 10.1016/j.anihpc.2010.01.009. [13] G. Devillanova and S. Solimini, Concentrations estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 7 (2002), 1257-1280. [14] W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336. [15] B. Esry, C. Greene, J. Burke Jr. and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. [16] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1983. doi: 10.1007/978-3-642-61798-0. [17] Q. Han and F. Lin, Elliptic Paritial Differential Equations,$2^nd$edition, AMS, 2011. [18] P. L. Lions, The concentration compactness principle in the calculus of variations Parts Ⅰ and Ⅱ, Ann. Ins. H. Poincaré, 1 (1984), 109-145, 223-283. [19] T. Lin and J. Wei, Ground state of$N$couple nonlinear Schrödinger equations in$\mathbb{R}^n, n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. [20] T. C. Lin and J. Wei, Spike in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincar Ana. Non Linaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [21] C. Maniscalco, Multiple solutions for semilinear elliptic problems in$\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110. [22] Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235. [23] E. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform holder bounds for nonlinear Schrödinger system with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309. [24] R. Palais, Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132. doi: 10.1016/0040-9383(66)90013-9. [25] S. Peng and Z. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger system, Arch. Rat. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0. [26] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [27] Y. Rudyak and F. Schlenk, Lusternik-Schnirelmann theory for fixed points of maps, Topological Methods in Nonlinear Analysis, 21 (2003), 171-194. doi: 10.12775/TMNA.2003.011. [28] S. Solimini, Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453. doi: 10.1007/BF01171757. [29] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. [30] S. Terracini and G. Verzini, Multiple phase in$k-$mixture of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y. [31] J. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equation, Rend. Lincei Mat. Appl., 18 (2007), 279-293. doi: 10.4171/RLM/495. [32] J. Wei and T. Weth, Radial solutions and phase sepration in a system of two coupled systems of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9. [33] J. 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. [34] C. Zelati and P. Robinowitz, Homoclinic type solutions for a semilinear elliptic PDE on$\mathbb{R}^N$, Comm. Pure. Appl. Math., 10 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  [1] Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 [2] Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure and Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965 [3] Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242 [4] Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109 [5] Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265 [6] Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104 [7] Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094 [8] Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004 [9] Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427 [10] Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4705-4736. doi: 10.3934/dcds.2021054 [11] Lixi Wen, Wen Zhang. Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022109 [12] Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003 [13] Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous$p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51 [14] Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 [15] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [16] Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3723-3744. doi: 10.3934/cpaa.2021128 [17] Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular$ \phi $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015 [18] Petru Jebelean. Infinitely many solutions for ordinary$p\$-Laplacian systems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267 [19] Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [20] Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7

2020 Impact Factor: 1.916