In this paper, we consider the following Chern-Simons-Schrödinger system where $u \in H^{1}(\mathbb{R}^{2})$ , $p > 4$ , $A_{\alpha }: \mathbb{R}^{2} \rightarrow \mathbb{R}$ are the components of the gauge potential, $N: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a neutral scalar field, $V(x)$ is a periodic potential function, the parameters $\kappa , q>0$ represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and $e>0$ is the coupling constant. We prove that system $(P)$ has a nontrivial solution by using a new infinite-dimensional linking theorem.
