We study a gradient system in \(\mathbb {R}^{2}\) given by \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \left[ 1+\mu W_1(x)\right] u=Q_u(u,v)& \text{ in } \ \ \mathbb {R}^2,\\ & \\ -\Delta v + \left[ 1+\mu W_2(x)\right] v=Q_v(u,v)& \text{ in } \ \ \mathbb {R}^2. \\ & \\ \end{array} \right. \end{aligned}\) The nonlinearity Q has exponential subcritical or critical growth. We prove the existence of a positive weak solution with minimal energy \((u_{\mu },v_{\mu }) \in L^{\infty }(\mathbb {R}^{2})\cap C^{1,\iota }_{loc}(\mathbb {R}^{2})\times L^{\infty }(\mathbb {R}^{2})\cap C^{1,\iota }_{loc}(\mathbb {R}^{2})\) , for some \(0<\iota <1\) . We also show a concentration result for this positive solution \((u_{\mu }, v_{\mu })\) as \(\mu \rightarrow + \infty \) .
