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

Multiple solutions for a nonlinear Schrödinger systems

Fengshuang Gao and  Yuxia Guo ,

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<p<\frac{2N}{N-2} = 2^{\ast} $
,
$ \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). $
Keywords: Nonlinear Schrödinger system, non-symmetry potential, infinitely many solutions.
Mathematics Subject Classification: Primary: 35B45, 35J25.
Citation: Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Devillanova and S. Solimini, Concentrations estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 7 (2002), 1257-1280.   Google Scholar

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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.  Google Scholar

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B. Esry, C. Greene, J. Burke Jr. and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.   Google Scholar

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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.  Google Scholar

[17]

Q. Han and F. Lin, Elliptic Paritial Differential Equations, $2^nd$ edition, AMS, 2011. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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C. Maniscalco, Multiple solutions for semilinear elliptic problems in $\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110.   Google Scholar

[22]

Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235.   Google Scholar

[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.  Google Scholar

[24]

R. Palais, Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132.  doi: 10.1016/0040-9383(66)90013-9.  Google Scholar

[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.  Google Scholar

[26]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[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.  Google Scholar

[28]

S. Solimini, Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453.  doi: 10.1007/BF01171757.  Google Scholar

[29]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. Adams, Sobolev Spaces, Academic press, New York-London, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

B. Esry, C. Greene, J. Burke Jr. and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.   Google Scholar

[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.  Google Scholar

[17]

Q. Han and F. Lin, Elliptic Paritial Differential Equations, $2^nd$ edition, AMS, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Maniscalco, Multiple solutions for semilinear elliptic problems in $\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110.   Google Scholar

[22]

Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235.   Google Scholar

[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.  Google Scholar

[24]

R. Palais, Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132.  doi: 10.1016/0040-9383(66)90013-9.  Google Scholar

[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.  Google Scholar

[26]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[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.  Google Scholar

[28]

S. Solimini, Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453.  doi: 10.1007/BF01171757.  Google Scholar

[29]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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