| الملخص: | In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations:
\begin{equation*}
\begin{cases}
-\Delta u+V_1(x)u =f_1(x,u)+\lambda(x) v,& x\in \mathbb{R}^2, \\
-\Delta v+V_2(x)v=f_2(x,v)+\lambda(x) u,& x\in \mathbb{R}^2,
\end{cases}
\end{equation*}
where $ \lambda, V_1, V_2\in C(\mathbb{R}^2,(0,+\infty))$ and $ f_1, f_2:\mathbb{R}^2\times \mathbb{R} \rightarrow \mathbb{R} $ have critical exponential growth in the sense of Trudinger–Moser inequality. The potentials $ V_1(x) $ and $ V_2(x) $ satisfy a condition involving the coupling term $ \lambda(x) $, namely $ 0<\lambda(x)\le\lambda_0\sqrt{V_1(x)V_2(x)}$. We use non-Nehari manifold, Lions's concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and $ L^q $-estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results.
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