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

Positive ground state solutions of a quadratically coupled schrödinger system

Zhanping Liang , Yuanmin Song and  Fuyi Li ,

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.
Keywords: Schrödinger system, quadratic nonlinearities, critical growth, ground state, ingenious hypotheses.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J50.
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.  Google Scholar

[2]

T. Bartsch, N. 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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

A. V. Buryak, P. Di Trapani, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. C. Yew, A. 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.  Google Scholar

[18]

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

[19]

L. Zhao, F. 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.  Google Scholar

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.  Google Scholar

[2]

T. Bartsch, N. 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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

A. V. Buryak, P. Di Trapani, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. C. Yew, A. 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.  Google Scholar

[18]

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

[19]

L. Zhao, F. 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.  Google Scholar

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