Solutions to Chern-Simons-Schrödinger systems with external potential

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system

$ \begin{equation} \left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right. \end{equation} $

with an external potential $ V(x) $, where $ D_{0} = \partial_{t}+i\lambda A_{0} $ and $ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $ for $ (x_1,x_2,t)\in \mathbb{R}^{2,1} $ are covariant derivatives, $ \lambda $ is the coupling number. Suppose that $ V(x) $ satisfies $ \lim_{|x|\to\infty}V(x) = +\infty $, we show for $ 2<p<4 $ that there exists $ \lambda^*>0 $ such that if $ 0<\lambda<\lambda^* $, problem (1) has two nontrivial static solutions $ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $. Moreover, there also exists $ \tilde\lambda>0 $ such that if $ \lambda>\tilde\lambda $, problem (1) has no nontrivial solutions. While for $ p>4 $ we assume in addition that $ x\cdot \nabla V(x)\geq 0 $, then problem (1) admits a mountain pass solution for all $ \lambda>0 $.