Normalized Solutions for the Nonlinear Biharmonic Choquard Equation with Hardy–Littlewood–Sobolev Upper Critical Exponent

In this paper, we investigate the following nonlinear Biharmonic Choquard equation with Hardy–Littlewood–Sobolev upper critical exponent \[ \begin{cases} \Delta^{2} u = \lambda u + \mu(I_{\alpha} * |u|^{p}) |u|^{p-2} u + (I_{\alpha} * |u|^{4^{*}_{\alpha}}) |u|^{4^{*}_{\alpha}-2} u, &x \in \mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = \rho, \end{cases} \] where $N \geq 5$, $\mu \gt 0$, $\rho \gt 0$, $0 \lt \alpha \lt N$, $(N+\alpha)/N \lt p \lt 4^{*}_{\alpha} := (N+\alpha)/(N-4)$, $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier and $I_{\alpha}$ is the Riesz potential. Under different assumptions on $p$, we show the existence of normalized solutions by variational method and Pohožaev manifold. Moreover, we describe the behavior of the solutions as $\mu \to 0$.