Ground states for a system of nonlinear Schrödinger equations with singular potentials

Peng Chen; Xianhua Tang

doi:10.3934/dcds.2022088

Article Contents

2022, Volume 42, Issue 10: 5105-5136. Doi: 10.3934/dcds.2022088

This issue Previous Article Oriented and orbital shadowing for vector fields Next Article Closed relations with non-zero entropy that generate no periodic points

Ground states for a system of nonlinear Schrödinger equations with singular potentials

Peng Chen^1,2, , and
Xianhua Tang^3,

1.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, Hubei 443002, China
2.
College of Science, China Three Gorges University, Yichang, Hubei 443002, China
3.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

^*Corresponding author: Peng Chen
^*Corresponding author: Peng Chen

Received: January 31, 2022

Revised: April 30, 2022

Early access: July 2022

Published: September 2022

The first author is supported by Natural Science Foundation of Hubei Province of China (2021CFB473)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions.

Keywords:

Mathematics Subject Classification: Primary: 37J45; Secondary: 58E05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.
[2]	D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. doi: 10.1016/j.jde.2004.03.005.
[3]	D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problem with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424-434. doi: 10.1016/S0022-0396(03)00118-9.
[4]	D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. doi: 10.1090/S0002-9939-02-06729-1.
[5]	S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D., 196 (2004), 341-361. doi: 10.1016/j.physd.2004.06.002.
[6]	P. Chen, X. Tang and L. Zhang, Non-Nehari manifold method for Hamiltonian elliptic system with Hardy potential: Existence and asymptotic properties of ground state solution, J. Geom. Anal., 32 (2022), Paper No. 46, 39 pp. doi: 10.1007/s12220-021-00739-5.
[7]	Z. Chen and W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. doi: 10.1016/j.jde.2011.09.042.
[8]	Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 1376-1398. doi: 10.1016/j.jde.2012.05.009.
[9]	V. Felli, On the existence of ground state solutions to nonlinear Schrödinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math., 108 (2009), 189-217. doi: 10.1007/s11854-009-0023-2.
[10]	V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019.
[11]	V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. doi: 10.1080/03605300500394439.
[12]	Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202. doi: 10.1016/j.jde.2015.11.006.
[13]	Y. Guo, S. Li, J. Wei and X. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅰ: Existence and uniqueness, J. Funct. Anal., 276 (2019), 183-230. doi: 10.1016/j.jfa.2018.09.015.
[14]	Y. Guo, S. Li, J. Wei and X. Zeng, Ground States of two-component attractive Bose-Einstein condensates Ⅱ: Semi-trivial limit behavior, Trans. Amer. Math. Soc., 371 (2019), 6903-6948. doi: 10.1090/tran/7540.
[15]	G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.
[16]	T.-C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.
[17]	T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n \leq 3$, Commu. Math. Phys., 277 (2008), 573-576. doi: 10.1007/s00220-007-0365-5.
[18]	S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9. doi: 10.1007/s00526-011-0447-2.
[19]	L. A. Maia, E. 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.
[20]	B. Malomed, Multi-component Bose-Einstein condensates: Theory, In Emergent Nonlinear Phenomena in Bose-Einstein Condensates, Springer-Verlag, Berlin, 45 (2008), 287–305. doi: 10.1007/978-3-540-73591-5_15.
[21]	J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Commu. Part. Diff. Eq., 41 (2016), 1426-1440. doi: 10.1080/03605302.2016.1209520.
[22]	E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71. doi: 10.4171/JEMS/103.
[23]	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.
[24]	A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570. doi: 10.1090/S0002-9939-08-09484-7.
[25]	S. Peng and Z. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.
[26]	M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. Ⅳ, Academic Press, New York, 1978.
[27]	D. Reiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. doi: 10.1016/S0022-0396(02)00178-X.
[28]	Ch. Rüegg, et al., Bose-Einstein condensation of the triple states in the magnetic insulator TICuCI3, Nature, 423 (2003), 62-65. doi: 10.1038/nature01617.
[29]	B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.
[30]	B. Sirakov, Least-energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[31]	D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9.
[32]	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.
[33]	X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math., 18 (2014), 1957-1979. doi: 10.11650/tjm.18.2014.3541.
[34]	X. H. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 55 (2017), Paper No. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.
[35]	X. Tang and X. Lin, Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems, Sci. China Math., 63 (2020), 113-134. doi: 10.1007/s11425-017-9332-3.
[36]	X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differ. Equ., 268 (2020), 4663-4690. doi: 10.1016/j.jde.2019.10.041.
[37]	J. Wei and Y. Wu, Ground states of nonlinear Schrödinger systems with mixed couplings, J. Math. Pure. Appl., 141 (2020), 50-88. doi: 10.1016/j.matpur.2020.07.012.
[38]	Y. Wu, Ground states of a $K$-component critical system with linear and nonlinear couplings: The attractive case, Adv. Nonlinear Stud., 19 (2019), 595-623. doi: 10.1515/ans-2019-2049.
[39]	M. Yang, W. Chen and Y. Ding, Solutions of a class of Hamiltonian elliptic systems in $\mathbb{R}^N$, J. Math. Anal. Appl., 352 (2010), 338-349. doi: 10.1016/j.jmaa.2009.07.052.
[40]	J. Zhang, X. Tang and W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1-10. doi: 10.1016/j.na.2013.07.027.
[41]	J. Zhang, W. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Disc. Contin. Dyn. Syst., 37 (2017), 4565-4583. doi: 10.3934/dcds.2017195.
[42]	J. Zhang, W. Zhang and X. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599-622. doi: 10.3934/cpaa.2016.15.599.
[43]	F. Zhao and Y. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials, J. Differential Equations, 249 (2010), 2964-2985. doi: 10.1016/j.jde.2010.09.014.