Normalized solutions for fractional Choquard equation with critical growth on bounded domain

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$\{\begin{array}{l}(-\mathrm{\Delta }{)}^{s}u=\lambda u+\alpha \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\left(\underset{\mathrm{\Omega }}{\int }\frac{\mathrm{\mid }u(y){\mathrm{\mid }}^{2_{\mu ,s}^{\ast }}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}dy\right)\mathrm{\mid }u{\mathrm{\mid }}^{2_{\mu ,s}^{\ast }-2}u\text{in}\mathrm{\Omega },\\ u>0\text{in}\mathrm{\Omega },\\ u=0\text{in}{\mathbb{R}}^N\mathrm{\backslash }\mathrm{\Omega },\\ {\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=d,\end{array}$$ where, $s\in (0,1),N>2s$, $\alpha \in \mathbb{R}$, $d>0$, $2<p<2_s^{\ast }:=\frac{2N}{N-2s}$ and $2_{\mu ,s}^{\ast }:=\frac{2N-\mu }{N-2s}$ represents fractional Hardy-Littlewood-Sobolev critical exponent. Using the minimization technique over an appropriate set and the uniform mountain pass theorem, we prove the existence of first and second solutions, respectively.