Existence of ground state for coupled system of biharmonic Schrödinger equations

Yanhua Wang; Min Liu; Gongming Wei; Yanhua Wang; Min Liu; Gongming Wei

doi:10.3934/math.2022206

AIMS Mathematics

2022, Volume 7, Issue 3: 3719-3730. doi: 10.3934/math.2022206

Previous Article Next Article

Research article

Existence of ground state for coupled system of biharmonic Schrödinger equations

1.
School of Mathematics, Shanghai Key Laboratory of Financial Information Technology, Shanghai University of Finance and Economics, Shanghai 200433, China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received: 26 September 2021 Accepted: 17 November 2021 Published: 07 December 2021
MSC : 35J35, 35J50, 35Q55, 47J35

In this paper we consider the following system of coupled biharmonic Schrödinger equations

$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $

where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} > 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.
- system of Schrödinger equations,
- ground state solution,
- Nehari manifold,
- concentration-compactness principle
Citation: Yanhua Wang, Min Liu, Gongming Wei. Existence of ground state for coupled system of biharmonic Schrödinger equations[J]. AIMS Mathematics, 2022, 7(3): 3719-3730. doi: 10.3934/math.2022206

Related Papers:

Abstract

In this paper we consider the following system of coupled biharmonic Schrödinger equations

$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $

where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} > 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.

References

[1]	R. A. Adams, J. F. Fournier, Sobolev spaces, Pure Appl. Math. (Amst), 2003. doi: 10.1093/oso/9780198812050.003.0005. doi: 10.1093/oso/9780198812050.003.0005
[2]	A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci., 342 (2006), 453–458. doi: 10.1016/j.crma.2006.01.024. doi: 10.1016/j.crma.2006.01.024
[3]	A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67–82. doi: 10.1112/jlms/jdl020. doi: 10.1112/jlms/jdl020
[4]	A. Ambrosetti, E. Colorado, D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. PDEs, 30 (2007), 85–112. doi: 10.1007/s00526-006-0079-0. doi: 10.1007/s00526-006-0079-0
[5]	P. Alvarez-Caudevilla, E. Colorado, R. Fabelo, A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations, J. Math. Phys., 58 (2017), 111503. doi: 10.1063/1.5010682. doi: 10.1063/1.5010682
[6]	H. Brezis, Functional analysis, sobolev spaces and partial differential equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.
[7]	E. Colorado, Existence results for some systems of coupled fractional nonlinear Schrödinger equations, Contemp. Math., 135–150. doi: 10.1090/conm/595/11805.
[8]	X. Duan, G. Wei, H. Yang, Positive solutions and infinitely many solutons for a weakly coupled system, Acta Math. Sci., 40B (2020), 1585–1601. doi: 10.1007/s10473-020-0523-9. doi: 10.1007/s10473-020-0523-9
[9]	X. Duan, G. Wei, H. Yang, Ground states for a coupled Schrödinger system with general nonlinearities, Bound. Value Probl., 22 (2020). doi: 10.1186/s13661-020-01331-6. doi: 10.1186/s13661-020-01331-6
[10]	X. Duan, G. Wei, H. Yang, Ground states for a fractional Schrödinger-Poisson system involving Hardy potentials, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1778672. doi: 10.1080/00036811.2020.1778672
[11]	G. Fibich, B. Ilan, G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437–1462. doi: 10.1137/s0036139901387241. doi: 10.1137/s0036139901387241
[12]	Y. Guo, J. Liu, Liouville type theorem for positive solutions of elliiptic systems in $R.{N}$, Comm. PDEs, 33 (2008), 263–284. doi: 10.1080/03605300701257476. doi: 10.1080/03605300701257476
[13]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336–1339. doi: 10.1103/physreve.53.r1336. doi: 10.1103/physreve.53.r1336
[14]	V. I. Karpman, A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger type equation with higher-order dispersion, Phys. D, 144 (2000), 194–210. doi: 10.1016/s0167-2789(00)00078-6. doi: 10.1016/s0167-2789(00)00078-6
[15]	C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb R.n$, Comment. Math. Helv., 73 (1998), 206–231. doi: 10.1007/s000140050052. doi: 10.1007/s000140050052
[16]	P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334. doi: 10.1016/0022-1236(82)90072-6. doi: 10.1016/0022-1236(82)90072-6
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 and part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/s0294-1449(16)30428-0 and 10.1016/s0294-1449(16)30422-x.
[18]	T. C. Lin, J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $R.{N}$, $N\leq3$, Comm. Math. Phys., 255 (2005), 629–653. doi: 10.1007/s00220-005-1313-x. doi: 10.1007/s00220-005-1313-x
[19]	L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equations, 229 (2006), 743–767. doi: 10.1016/j.jde.2006.07.002. doi: 10.1016/j.jde.2006.07.002
[20]	L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa CI. Sci., 13 (1959), 115–162. doi: 10.1007/978-3-642-10926-3-1. doi: 10.1007/978-3-642-10926-3-1
[21]	P. Quittner, P. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Commun. Math. Phys., 311 (2012), 1–19. doi: 10.1007/s00220-012-1440-0. doi: 10.1007/s00220-012-1440-0
[22]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in ${\mathbb{R}}.{N}$, Commun. Math. Phys., 271 (2007), 199–221. doi: 10.1007/s00220-006-0179-x. doi: 10.1007/s00220-006-0179-x
[23]	H. Tavares, S. Terracini, G. Verzini, T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems, Commun. Part. Diff. Eq., 36 (2011), 1988–2010. doi: 10.1080/03605302.2011.574244. doi: 10.1080/03605302.2011.574244
[24]	M. Willem, Minimax Theorems, Birkhauser Bostonda, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.
[25]	G. Wei, X. Duan, On existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials, Rocky Mountain J. Math., 48 (2018), 1647–1683. doi: 10.1216/rmj-2018-48-5-1647. doi: 10.1216/rmj-2018-48-5-1647
[26]	J. W. Yang, Y. T. Gao, Q. M. Wang, Q. C. Su, Y. C. Feng, X. Yu, Bilinear forms and soliton solutions for a fourth order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein, Phys. B, 481 (2016), 148–155. doi: 10.1016/j.physb.2015.10.025. doi: 10.1016/j.physb.2015.10.025

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)