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

Abstract

In this paper, we consider the existence of multi-Peak solutions to the nonlinear Chern-Simons-Schrödinger system $$ \begin{cases} -ihD_0\Psi-h^2(D_1D_1+D_2D_2)\Psi+V\Psi=|\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0= -\frac 12ih[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0= \frac 12ih[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1= -\frac12|\Psi|^2, \end{cases} \qquad\qquad (0.1) $$ where $p > 2$ and $V(x)$ is a non-radial potential. Our main result states that for every positive integer $k$, we can find $h_0>0$ such that for $0 < h < h_0$, problem (0.1) has a nontrivial static solution $(\Psi_h, A_0^h, A_1^h,A_2^h)$. Moreover, $\Psi_h$ has $k$ positive peaks, which tend to the local maximum point of $V(x)$ as $h\to 0^+$.