Normalized solutions for the Choquard equations with critical nonlinearities

This study is concerned with the existence of normalized solutions for the Choquard equations with critical nonlinearities −Δu+λu=f(u)+(Iα∗∣u∣2α*)∣u∣2α*−2u,inRN,∫RN∣u∣2dx=a2,\left\{\begin{array}{l}-\Delta u+\lambda u=f\left(u)+\left({I}_{\alpha }\ast {| u| }^{{2}_{\alpha }^{* }}){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where N>2N\gt 2, α∈(0,N)\alpha \in \left(0,N), a>0a\gt 0, and Iα(x){I}_{\alpha }\left(x) is the Riesz potential given by Iα(x)=Aα∣x∣N−αwithAα=ΓN−α22απN2Γα2,{I}_{\alpha }\left(x)=\frac{{A}_{\alpha }}{{| x| }^{N-\alpha }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\alpha }=\frac{\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{N-\alpha }{2}\right)}{{2}^{\alpha }{\pi }^{\tfrac{N}{2}}\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{\alpha }{2}\right)}, and 2α*=N+αN−2{2}_{\alpha }^{* }=\frac{N+\alpha }{N-2} is the Hardy-Littlewood-Sobolev critical exponent and ff is a subcritical nonlinearity. In the case that ff is L2{L}^{2}-supercritical growth, by means of the Pohozaev manifold method and mountain pass theorem, we obtain a couple of the normalized solution; while in the case f(u)=μ∣u∣q−2uf\left(u)=\mu {| u| }^{q-2}u with 20\mu \gt 0 a parameter, we employ the truncation technique and the genus theory to prove the multiplicity of normalized solutions.