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# Solutions to Chern-Simons-Schrödinger systems with external potential

• * Corresponding author: Jianfu Yang
• 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 $\lambda>0$.

Mathematics Subject Classification: Primary: 35J50; Secondary: 35J10.

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