The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent

In this article we are concern for the following Choquard equation $$-\mathrm{\Delta }u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\left({\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }u(y){\mathrm{\mid }}^{2_{\mu }^{\ast }}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}dy\right)\mathrm{\mid }u{\mathrm{\mid }}^{2_{\mu }^{\ast }-2}u\text{in}\mathrm{\Omega },u=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is an open bounded set with continuous boundary in ${\mathbb{R}}^N(N\ge 3)$, $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ and $q \in [2,2^)$where$2^=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\mathrm{\Omega }$. Indeed, we prove if $\lambda <{\lambda }_1$ then problem has ${\text{cat}}_{\mathrm{\Omega }}(\mathrm{\Omega })$ positive solutions whenever $q\in [2,2^{\ast })$ and $N>3$ or $4<q<6$ and $N=3$.