# 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 & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
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