Multiple Normalized Solutions for Biharmonic Choquard Equation with Hardy–Littlewood–Sobolev Upper Critical and Combined Nonlinearities

Abstract

In this paper, we study multiple normalized solutions for biharmonic Choquard equation with Hardy–Littlewood–Sobolev upper critical and combined nonlinearities

$$\begin{aligned} \Delta ^2u=\lambda u+\mu |u|^{q-2}u+(I_\alpha *|u|^{4_\alpha ^*})|u|^{4_\alpha ^*-2}u~\text {in}~{\mathbb {R}}^N,~~\int _{{\mathbb {R}}^N}|u|^2dx=a>0, \end{aligned}$$

where $N\ge 5$, $\mu >0$, $2<q<2+8/N$, $\alpha \in (0,N)$, $I_\alpha$ is the Riesz potential and $4_\alpha ^*=\frac{N+\alpha }{N-4}$. We first show the existence of normalized ground state solutions, and all ground states correspond to local minima of the associated energy functional. Next, when $N\ge 9$, we further assume that $4_\alpha ^*\ge 2$, there exist a class of solutions which are not ground states and are located at a mountain-pass level of the energy functional. This study may be the first contribution regarding existence of multiple normalized solutions for biharmonic Choquard equation.