May  2017, 16(3): 999-1012. doi: 10.3934/cpaa.2017048

Positive ground state solutions of a quadratically coupled schrödinger system

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China

* Corresponding author

Received  October 2016 Revised  December 2016 Published  February 2017

Fund Project: This work is supported by National Natural Science Foundation of China (Grant Nos. 11571209, 11671239)

In this paper, we study the following quadratically coupled Schrödinger system:
$\begin{equation*}\left\{\begin{array}{ll}-\Delta u+\lambda_1u=\mu_1u^2+2\alpha uv+\gamma v^2, & \mbox{in }\Omega,\\-\Delta v+\lambda_2v=\mu_2v^2+2\gamma uv+\alpha u^2, & \mbox{in }\Omega,\\u=v=0, & \mbox{on }\partial\Omega,\end{array}\right.\end{equation*}$
where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) < \lambda_1, \lambda_2 < 0, \mu_1, \mu_2, \alpha, \gamma>0$, and $\lambda (\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. The main difficulty to investigate this kind of equations is caused by the fact that all the quadratic nonlinearities, including the coupling terms, are of critical growth. By the methods used in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in $\mathbb{R}^4$, J. Math. Anal. Appl., 430(2):950-970, 2015], the existence of positive ground state solutions of the system is established with more ingenious hypotheses.
Citation: 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
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]

T. BartschN. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.

[3]

T. BartschZ.-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.

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[5]

A. V. BuryakP. Di TrapaniD. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235. doi: 10.1016/S0370-1573(02)00196-5.

[6]

A. V. Buryak and Y. S. Kivshar, Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412. doi: 10.1016/0375-9601(94)00989-3.

[7]

Z. Chen and W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107. doi: 10.1016/j.jfa.2012.01.001.

[8]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8.

[9]

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

[10]

Z. Chen and W. Zou, Standing waves for linearly coupled Schrödinger equations with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 429-447. doi: 10.1016/j.anihpc.2013.04.003.

[11]

E. N. DancerJ. 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.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. doi: 10.1007/978-3-642-61798-0.

[13]

Z. Guo, Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in ℝ4, J. Math. Anal. Appl., 430 (2015), 950-970. doi: 10.1016/j.jmaa.2015.05.037.

[14]

C. T.-Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in ℝn. n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[15]

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.

[16]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications. 24, Birkhäuser Boston, Inc., Boston, MA, (1996). doi: 10.1007/978-1-4612-4146-1.

[17]

A. C. YewA. R. Champneys and P. J. McKenna, Multiple solitary waves due to secondharmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52. doi: 10.1007/s003329900063.

[18]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

[19]

L. ZhaoF. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691. doi: 10.1007/s00526-015-0879-1.

show all references

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]

T. BartschN. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.

[3]

T. BartschZ.-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.

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[5]

A. V. BuryakP. Di TrapaniD. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235. doi: 10.1016/S0370-1573(02)00196-5.

[6]

A. V. Buryak and Y. S. Kivshar, Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412. doi: 10.1016/0375-9601(94)00989-3.

[7]

Z. Chen and W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107. doi: 10.1016/j.jfa.2012.01.001.

[8]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8.

[9]

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

[10]

Z. Chen and W. Zou, Standing waves for linearly coupled Schrödinger equations with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 429-447. doi: 10.1016/j.anihpc.2013.04.003.

[11]

E. N. DancerJ. 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.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. doi: 10.1007/978-3-642-61798-0.

[13]

Z. Guo, Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in ℝ4, J. Math. Anal. Appl., 430 (2015), 950-970. doi: 10.1016/j.jmaa.2015.05.037.

[14]

C. T.-Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in ℝn. n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[15]

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.

[16]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications. 24, Birkhäuser Boston, Inc., Boston, MA, (1996). doi: 10.1007/978-1-4612-4146-1.

[17]

A. C. YewA. R. Champneys and P. J. McKenna, Multiple solitary waves due to secondharmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52. doi: 10.1007/s003329900063.

[18]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

[19]

L. ZhaoF. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691. doi: 10.1007/s00526-015-0879-1.

[1]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[2]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

[3]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[4]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[5]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079

[6]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026

[7]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025

[8]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[9]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104

[10]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2018329

[11]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417

[12]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[13]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005

[14]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[15]

Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040

[16]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214

[17]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

[18]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[19]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019131

[20]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (14)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]