Summary: We establish the existence and concentration of solutions of a class of nonlinear Schrödinger equation \[ - \varepsilon^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma }, u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right), \] where \(2 < p < {\infty}, \alpha_0 > 0, 0 < \gamma < 2\). When the potential function \(V (x)\) decays at infinity like \((1 + |x|)-\alpha\) with \(0 < \alpha {\leq} 2\) and \(K(x) > 0\) are permitted to be unbounded under some necessary restrictions, we show that a positive \(H^1(\mathbb{R}^2)\)-solution \(u_\varepsilon\) exists if it is assumed that the corresponding ground energy function \(G(\xi)\) of the nonlinear Schrödinger equation \(- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }\) has local minimum points. Furthermore, the concentration property of \(u_\epsilon\) is also established as \(\varepsilon\) tends to zero.