May 2019, 18(3): 1261-1280. doi: 10.3934/cpaa.2019061

Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials

1. 

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

2. 

School of Traffic and Transportation Engineering, Central South University, Changsha, 410075 Hunan, China

* Corresponding author

Received  May 2018 Revised  September 2018 Published  November 2018

Fund Project: This work is partially supported by the National Natural Science Foundation of China (Nos.: 11801574, 11571370, 11501190) of China

In this paper we study a class of weakly coupled Schrödinger system arising in several branches of sciences, such as nonlinear optics and Bose-Einstein condensates. Instead of the well known super-quadratic condition that $\lim_{|z|\to∞}\frac{F(x,z)}{|z|^2} = ∞$ uniformly in $x$, we introduce a new local super-quadratic condition that allows the nonlinearity $F$ to be super-quadratic at some $x∈ \mathbb{R}^N$ and asymptotically quadratic at other $x∈ \mathbb{R}^N$. Employing some analytical skills and using the variational method, we prove some results about the existence of ground states for the system with periodic or non-periodic potentials. In particular, any nontrivial solutions are continuous and decay to zero exponentially as $|x| \to ∞$. Our main results extend and improve some recent ones in the literature.

Citation: Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061
References:
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A. AmbrosettiG. 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.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[3]

T. BartschA. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.

[4]

T. Bartsch, Z.-Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Handbook of Differential Equations-Stationary Partial Differential Equations (eds. M. Chipot and P. Quittner), vol. 2, Elsevier, 2005, pp. 1-5 (Chapter 1).

[5]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113.

[6]

G. W. Chen and S. W. Ma, Asymptotically or super linear cooperative elliptic systems in the whole space, Sci. China Math., 56 (2013), 1181-1194. doi: 10.1007/s11425-013-4567-3.

[7]

G. W. Chen and S. W. Ma, Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms, Calc. Var., 49 (2014), 271-286. doi: 10.1007/s00526-012-0581-5.

[8]

G. W. Chen and S. W. Ma, Nonexistence and multiplicity of solutions for nonlinear elliptic systems of $ \mathbb{R}^N$, Nonlinear Anal.-Real World Appl., 36 (2017), 233-248. doi: 10.1016/j.nonrwa.2017.01.012.

[9]

R. Cipolatti and W. Zumpichiatti, On the existence and regularity of ground states for a nonlinear system of coupled Schrödinger equations in $ \mathbb{R}^N$, Comput. Appl. Math., 18 (1999), 15-29.

[10]

D. G. Costa, On a Class of Elliptic Systems in $ \mathbb{R}^N$, Electron. J. Differential Equations, 7 (1994), 1-14.

[11]

D. G. Costa and C. A. Magalhães, A variational approach to subquadratic perturbations of elliptic systems, J. Differential Equations, 111 (1994), 103-122. doi: 10.1006/jdeq.1994.1077.

[12]

Y. H. Ding, Varitional Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2008. doi: 10.1142/9789812709639.

[13]

Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163. doi: 10.1016/j.jde.2005.03.011.

[14]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.

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Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.

[16]

B. D. EsryC. H. GreeneJ. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.

[17]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, Oxford, 1995. doi: 10.1007/BF00994627.

[18]

M. N. Islam, Ultrafast Fiber Switching Devices and Systems, Cambridge University Press, New York, 1992.

[19]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[20]

G. Li and A. Szulkin, An asymptotically periodic equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[21]

L. Li and C-L. Tang, Infinitely many solutions for resonance elliptic systems, C. R. Acad. Sci. Paris, Ser. I, 353 (2015), 35-40. doi: 10.1016/j.crma.2014.10.010.

[22]

Z. L. Liu and Z-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572. doi: 10.1515/ans-2004-0411.

[23]

L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008.

[24]

L. A. MaiaE. 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.

[25]

S. W. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal., 73 (2010), 3856-3872. doi: 10.1016/j.na.2010.08.013.

[26]

J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Commun. Partial Differ. Equ., 41 (2016), 1426-1440. doi: 10.1080/03605302.2016.1209520.

[27]

C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quant. Electron, 23 (1987), 174-176.

[28]

A. M. Molchanov, On the discreteness of the spectrum conditions for self-adjoint differential equations of the second order, Trudy Mosk. Matem. Obshchestva, 2 (1953), 169-199 (in Russian).

[29]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.

[30]

A. Pankov, On decay of solutions to nolinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570. doi: 10.1090/S0002-9939-08-09484-7.

[31]

D. D. Qin and X. H. Tang, Solutions on asymptotically periodic elliptic system with new conditions, Results. Math., 70 (2016), 539-565. doi: 10.1007/s00025-015-0491-x.

[32]

D. D. QinY. B. He and X. H. Tang, Ground and bound states for non-linear Schrödinger systems with indefinite linear terms, Complex Var. Elliptic Equ., 62 (2017), 1758-1781. doi: 10.1080/17476933.2017.1281256.

[33]

D. D. QinJ. Chen and X. H. Tang, Existence and non-existence of nontrivial solutions for Schrödinger systems via Nehari-Pohozaev manifold, Comput. Math. Appl., 74 (2017), 3141-3160. doi: 10.1016/j.camwa.2017.08.010.

[34]

Q. F. Wu and D. D. Qin, Ground and bound states of periodic Schrödinger equations with super or asymptotically linear terms, Electronic Journal of Differential Equations, 25 (2018), 1-26.

[35]

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

[36]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅳ, Analysis of Operators, Academic Press, New York, 1978.

[37]

M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77 (1980), 482-492. doi: 10.1016/0022-247X(80)90242-5.

[38]

M. Schechter and W. M. Zou, Weak linking theorems and Schrödinger equations with critical Soblev exponent, ESAIM Contral Optim. Calc. Var., 9 (2003), 601-619 (electronic). doi: 10.1051/cocv:2003029.

[39]

B. Simon, Schrödinger semigroup, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.

[40]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.

[41]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1.

[42]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., (2018), DOI: 10.1007/s10884-018-9662-2.

[43]

E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.

[44]

J. Vélin and F. de Thélin, Existence and non-existence of nontrivial solutions for some nonlinear elliptic systems, Rev. Mat. Univ. Complutense Madrid, 6 (1993), 153-154.

[45]

J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.

[46]

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

show all references

References:
[1]

A. AmbrosettiG. 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.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[3]

T. BartschA. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.

[4]

T. Bartsch, Z.-Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Handbook of Differential Equations-Stationary Partial Differential Equations (eds. M. Chipot and P. Quittner), vol. 2, Elsevier, 2005, pp. 1-5 (Chapter 1).

[5]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Commun. Math. Phys., 96 (1984), 97-113.

[6]

G. W. Chen and S. W. Ma, Asymptotically or super linear cooperative elliptic systems in the whole space, Sci. China Math., 56 (2013), 1181-1194. doi: 10.1007/s11425-013-4567-3.

[7]

G. W. Chen and S. W. Ma, Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms, Calc. Var., 49 (2014), 271-286. doi: 10.1007/s00526-012-0581-5.

[8]

G. W. Chen and S. W. Ma, Nonexistence and multiplicity of solutions for nonlinear elliptic systems of $ \mathbb{R}^N$, Nonlinear Anal.-Real World Appl., 36 (2017), 233-248. doi: 10.1016/j.nonrwa.2017.01.012.

[9]

R. Cipolatti and W. Zumpichiatti, On the existence and regularity of ground states for a nonlinear system of coupled Schrödinger equations in $ \mathbb{R}^N$, Comput. Appl. Math., 18 (1999), 15-29.

[10]

D. G. Costa, On a Class of Elliptic Systems in $ \mathbb{R}^N$, Electron. J. Differential Equations, 7 (1994), 1-14.

[11]

D. G. Costa and C. A. Magalhães, A variational approach to subquadratic perturbations of elliptic systems, J. Differential Equations, 111 (1994), 103-122. doi: 10.1006/jdeq.1994.1077.

[12]

Y. H. Ding, Varitional Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2008. doi: 10.1142/9789812709639.

[13]

Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163. doi: 10.1016/j.jde.2005.03.011.

[14]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.

[15]

Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.

[16]

B. D. EsryC. H. GreeneJ. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.

[17]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, Oxford, 1995. doi: 10.1007/BF00994627.

[18]

M. N. Islam, Ultrafast Fiber Switching Devices and Systems, Cambridge University Press, New York, 1992.

[19]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[20]

G. Li and A. Szulkin, An asymptotically periodic equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[21]

L. Li and C-L. Tang, Infinitely many solutions for resonance elliptic systems, C. R. Acad. Sci. Paris, Ser. I, 353 (2015), 35-40. doi: 10.1016/j.crma.2014.10.010.

[22]

Z. L. Liu and Z-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572. doi: 10.1515/ans-2004-0411.

[23]

L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008.

[24]

L. A. MaiaE. 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.

[25]

S. W. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal., 73 (2010), 3856-3872. doi: 10.1016/j.na.2010.08.013.

[26]

J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Commun. Partial Differ. Equ., 41 (2016), 1426-1440. doi: 10.1080/03605302.2016.1209520.

[27]

C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quant. Electron, 23 (1987), 174-176.

[28]

A. M. Molchanov, On the discreteness of the spectrum conditions for self-adjoint differential equations of the second order, Trudy Mosk. Matem. Obshchestva, 2 (1953), 169-199 (in Russian).

[29]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.

[30]

A. Pankov, On decay of solutions to nolinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570. doi: 10.1090/S0002-9939-08-09484-7.

[31]

D. D. Qin and X. H. Tang, Solutions on asymptotically periodic elliptic system with new conditions, Results. Math., 70 (2016), 539-565. doi: 10.1007/s00025-015-0491-x.

[32]

D. D. QinY. B. He and X. H. Tang, Ground and bound states for non-linear Schrödinger systems with indefinite linear terms, Complex Var. Elliptic Equ., 62 (2017), 1758-1781. doi: 10.1080/17476933.2017.1281256.

[33]

D. D. QinJ. Chen and X. H. Tang, Existence and non-existence of nontrivial solutions for Schrödinger systems via Nehari-Pohozaev manifold, Comput. Math. Appl., 74 (2017), 3141-3160. doi: 10.1016/j.camwa.2017.08.010.

[34]

Q. F. Wu and D. D. Qin, Ground and bound states of periodic Schrödinger equations with super or asymptotically linear terms, Electronic Journal of Differential Equations, 25 (2018), 1-26.

[35]

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

[36]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅳ, Analysis of Operators, Academic Press, New York, 1978.

[37]

M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77 (1980), 482-492. doi: 10.1016/0022-247X(80)90242-5.

[38]

M. Schechter and W. M. Zou, Weak linking theorems and Schrödinger equations with critical Soblev exponent, ESAIM Contral Optim. Calc. Var., 9 (2003), 601-619 (electronic). doi: 10.1051/cocv:2003029.

[39]

B. Simon, Schrödinger semigroup, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.

[40]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.

[41]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1.

[42]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., (2018), DOI: 10.1007/s10884-018-9662-2.

[43]

E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.

[44]

J. Vélin and F. de Thélin, Existence and non-existence of nontrivial solutions for some nonlinear elliptic systems, Rev. Mat. Univ. Complutense Madrid, 6 (1993), 153-154.

[45]

J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.

[46]

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

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