# American Institute of Mathematical Sciences

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 and 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. 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. [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. [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. 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. [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. 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. [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. 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. [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. 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.

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##### 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. 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. [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. [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. 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. [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. 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. [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. 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. [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. 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.
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