Multi-Peak solutions to Chern-Simons-Schrödinger systems with non-radial potential

Abstract

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system \begin{equation}\label{eqabstr} \left\{\begin{array}{ll} -ihD_0Ψ-h^2(D_1D_1+D_2D_2)Ψ+VΨ=|Ψ|^{p-2}Ψ,\\ \partial_0A_1-\partial_1A_0=-\frac 12ih[\overlineΨD_2Ψ-Ψ\overline{D_2Ψ}],\\ \partial_0A_2-\partial_2A_0=\frac 12ih[\overlineΨD_1Ψ-Ψ\overline{D_1Ψ}],\\ \partial_1A_2-\partial_2A_1=-\frac12|Ψ|^2,\\ \end{array} \right. \end{equation} where $p>2$ and non-radial potential $V(x)$ satisfies some certain conditions. We show that for every positive integer $k$, there exists $h_0>0$ such that for $0<h<h_0$, problem \eqref{eqabstr} has a nontrivial static solution $({\Psi }_h,A_0^h,A_1^h,A_2^h)$. Moreover, ${\Psi }_h$ is a positive non-radial function with $k$ positive peaks, which approach to the local maximum point of $V(x)$ as $h\to 0^{+}$.