Multiple solutions for a nonlinear Schrödinger systems

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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 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.

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