This paper considers the following elliptic problem with logarithmic nonlinearity and Hardy-Littlewood-Sobolev critical exponent \( \left\{ \begin{aligned}&-\Delta u=\mu |u|^{p-2}u\ln |u|^2+\Big (\int _{\Omega }\frac{|u(y)|^{2_\alpha ^*}}{|x-y|^\alpha }\textrm{d}y\Big )|u|^{2_\alpha ^*-2}u,&\text{ in }\,\,\Omega ,\\&u=0,&\text{ on }\,\,\partial \Omega , \end{aligned} \right. \) where \(\Omega \) is a bounded domain of \(\mathbb {R}^N\,(N\ge 3)\) with smooth boundary. \(\mu >0\) , \(0<\alpha <N\) and \(2^*_{\alpha }=\frac{2N -\alpha }{N - 2}\) . Using the concentration compactness principle and Lusternik-Schnirelman theory, we establish a relation between the number of solutions of the above problem with the topology of \(\Omega \) when \(\max \{2,\frac{N}{N-2},\frac{4}{N-2}\}<p<2^*\) with \(2^*=\frac{2N}{N-2}\) .
