Normalized solutions to focusing Sobolev critical biharmonic Schrödinger equation with mixed dispersion

This paper is concerned with the following focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth: $$\{\begin{array}{l}{\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u-\lambda u-\mu \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u-\mathrm{\mid }u{\mathrm{\mid }}^{4^{\ast }-2}u=0in{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}{u}^{2}dx=c,\end{array}$$ where $N\ge 5$, $\mu ,c>0$, $2<p<4^{\ast }:=\frac{2N}{N-4}$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier. For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concentration-compactness principle and the genus theory presented by C.O. Alves et al. (Arxiv, (2021), doi: 2103.07940v2). Compared to the results of C.O. Alves et al. we obtain a more general result after removing the further assumptions given in (3.2) of their paper. In the case of $L^2$-supercritical perturbation ($2+\frac{8}{N}<p<4^{\ast }$), we explore the existence results of normalized solutions by applying the constrained variational methods and the mountain pass theorem. Moreover, we propose a novel method to address the effects of the dispersion term $\mathrm{\Delta }u$. This approach allows us to extend the recent results obtained by X. Chang et al. (Arxiv, (2023), doi: 2305.00327v1) to the mixed dispersion situation.