Boundary concentration of peak solutions for fractional Schr\"{o}dinger-Poisson system

The goal of this paper is to study the existence of peak solutions for the following fractional Schr"{o}dinger-Poisson system: \begin{eqnarray*} \left{ \arraycolsep=1.5pt \begin{array}{ll} \varepsilon^{2s}(-\Delta)^{s}u+u+\phi u=u^p,\ \ \ &\ \mbox{in}
\Omega,\[2mm] (-\Delta)^{s}\phi=u^2,\ \ \ &\ \mbox{in}\ \Omega,\[2mm] u=\phi=0,\ \ \ \ &\ \mbox{in}\ $\mathbb{R}$^N\setminus \Omega, \end{array} \right. \end{eqnarray*} where $s\in (0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with Lipschitz boundary, and $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian operator, $\epsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_{\epsilon },{\varphi }_{\epsilon })$ such that the peak of $u_{\epsilon }$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance ${\epsilon }^{2\mathrm{/}3}$ away from the boundary $\mathrm{\partial }\mathrm{\Omega }$ as $\epsilon$ tends to 0, some new estimates and analytic technique are used.