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\\text{ in } \\Omega, &\\\\ -\\Delta v+\\lambda_2v=\\mu_2v(e^{v^2}-1)+\\beta u\\left(e^{uv}-1\\right)\\text{ in } \\Omega, &\\\\ u,v\\in H^1_0(\\Omega) \\end{cases} \\end{equation} where $\\Omega$ is a bounded smooth domain, $\\lambda_1,\\lambda_2>-\\Lambda_1$ (the first eigenvalue of $(-\\Delta,H^1_0(\\Omega))$, $\\mu_1,\\mu_2>0$ and $\\beta$ is either positive (small or large) 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