Multi-Peak solutions to Chern-Simons-Schr\"odinger systems with non-radial potential

TL;DR

This paper proves the existence of multi-peak, non-radial static solutions to a nonlinear Chern-Simons-Schrödinger system with a non-radial potential, showing solutions concentrate near potential maxima as a parameter tends to zero.

Contribution

It establishes the existence of multi-peak solutions with non-radial symmetry for the Chern-Simons-Schrödinger system under certain potential conditions, a novel result in the field.

Findings

Existence of solutions with arbitrary number of peaks.

Solutions concentrate near local maxima of the potential.

Solutions are non-radial and positive with multiple peaks.

Abstract

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schr\"odinger system \begin{equation}\label{eqabstr} \left\{\begin{array}{ll} -ihD_0\Psi-h^2(D_1D_1+D_2D_2)\Psi+V\Psi=|\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0=-\frac 12ih[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0=\frac 12ih[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1=-\frac12|\Psi|^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 $h\to 0^+$.