Solutions to Strongly Indefinite Chern-Simons-Schrödinger Systems

Abstract

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.