American Institute of Mathematical Sciences

Advanced
November  2019, 12(7): 1905-1927. doi: 10.3934/dcdss.2019125

Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems

Guowei Dai 1, , Rushun Tian 2, and  Zhitao Zhang 3,4,,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
3. 

HLM, CEMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
4. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Zhitao Zhang

Received  September 2017 Revised  April 2018 Published  December 2018

Fund Project: Dedicated to the 70th birthday of Professor E. N. Dancer. Research supported by NSFC(11771428, 11871129, 11601353, 11325107) and Beijing Natural Science Foundation(1174013).

In this paper, we consider the following coupled elliptic system
$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $
Under symmetric assumptions
$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $
, we determine the number of
$ \gamma $
-bifurcations for each
$ \beta\in(-1, +\infty) $
, and study the behavior of global
$ \gamma $
-bifurcation branches in
$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $
. Moreover, several results for
$ \gamma = 0 $
, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].
Keywords: Bifurcation, Schrödinger systems, positive solutions.
Mathematics Subject Classification: 35Q55, 35B32, 58C40, 58E07.
Citation: Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125
References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4.   Google Scholar

[2]

J. C. Alexander and S. S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.  doi: 10.1007/BF00249970.  Google Scholar

[3]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C.R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[4]

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.  Google Scholar

[5]

A. Ambrosetti, E. Colorado and D. Ruiz, Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.  Google Scholar

[6]

T. Bartsch, E. N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207.   Google Scholar

[8]

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[9]

J. Belmonte-Beitia, V. M. Prez-Garca and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.  doi: 10.1007/s00332-008-9037-7.  Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.  doi: 10.1112/plms/s3-30.1.76.  Google Scholar

[13]

E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.  doi: 10.1016/j.jfa.2011.10.013.  Google Scholar

[14]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769.  doi: 10.1016/j.jde.2011.06.015.  Google Scholar

[15]

E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[16]

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. Poincaré Anal. Non. Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[17]

B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: from Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706.   Google Scholar

[18]

N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988.  Google Scholar

[19]

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

[20]

P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084.  Google Scholar

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[22]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.  doi: 10.1007/s00030-009-0017-x.  Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[24]

K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp. doi: 10.1063/1.4960046.  Google Scholar

[25]

T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[26]

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.  Google Scholar

[27]

M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.   Google Scholar

[28]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

C. Rüegg, N. Cavadini, A. Furrer, H.-U. Gdel, K. Krmer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[33]

R. Tian and Z. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.  doi: 10.1007/s11425-015-5028-y.  Google Scholar

[34]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721.   Google Scholar

[35]

J. 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.  Google Scholar

show all references

References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4.   Google Scholar

[2]

J. C. Alexander and S. S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.  doi: 10.1007/BF00249970.  Google Scholar

[3]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C.R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[4]

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.  Google Scholar

[5]

A. Ambrosetti, E. Colorado and D. Ruiz, Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.  Google Scholar

[6]

T. Bartsch, E. N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207.   Google Scholar

[8]

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[9]

J. Belmonte-Beitia, V. M. Prez-Garca and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.  doi: 10.1007/s00332-008-9037-7.  Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.  doi: 10.1112/plms/s3-30.1.76.  Google Scholar

[13]

E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.  doi: 10.1016/j.jfa.2011.10.013.  Google Scholar

[14]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769.  doi: 10.1016/j.jde.2011.06.015.  Google Scholar

[15]

E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[16]

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. Poincaré Anal. Non. Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[17]

B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: from Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706.   Google Scholar

[18]

N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988.  Google Scholar

[19]

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

[20]

P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084.  Google Scholar

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[22]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.  doi: 10.1007/s00030-009-0017-x.  Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[24]

K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp. doi: 10.1063/1.4960046.  Google Scholar

[25]

T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[26]

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.  Google Scholar

[27]

M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.   Google Scholar

[28]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

C. Rüegg, N. Cavadini, A. Furrer, H.-U. Gdel, K. Krmer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[33]

R. Tian and Z. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.  doi: 10.1007/s11425-015-5028-y.  Google Scholar

[34]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721.   Google Scholar

[35]

J. 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.  Google Scholar

Figure 1.  Schematic diagrams of global $ \gamma $-bifurcations for $ \beta\in[\beta_{k+1}, \beta_k) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Schematic diagrams of β-bifurcation branches
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Global bifurcation diagram in the symmetric case µ1 = µ2 = 1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461
[2]

Hongyu Ye. Positive solutions for critically coupled Schrödinger systems with attractive interactions. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 485-507. doi: 10.3934/dcds.2018022
[3]

Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138
[4]

Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145
[5]

Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
[6]

Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033
[7]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4705-4736. doi: 10.3934/dcds.2021054
[8]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436
[9]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[10]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
[11]

Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077
[12]

Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003
[13]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
[14]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
[15]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294
[16]

Tai-Chia Lin, Tsung-Fang Wu. Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110
[17]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[18]

Eduard Toon, Pedro Ubilla. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5831-5843. doi: 10.3934/dcds.2020248
[19]

Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021214
[20]

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

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (262)
  • HTML views (660)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy