Existence and asymptotical behavior of normalized solutions to focusing biharmonic HLS upper critical Hartree equation with a local perturbation

Abstract: This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} {\Delta}^2u-\lambda u-\mu|u|^{p-2}u-(I_\alpha*|u|^{4^_\alpha})|u|^{4^_\alpha-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $0<\alpha<N$, $N \geq 5$, $\mu,c\>0$, $2+\frac{8}{N}=:\bar{p}\leq p<4^*:=\frac{2N}{N-4}$, $4^*_\alpha:=\frac{N+\alpha}{N-4}$, $\lambda \in \mathbb{R}$ is a Lagrange multiplier and $I_\alpha$ is the Riesz potential. Choosing an appropriate testing function, one can derive some reasonable estimate on the mountain pass level. Based on this point, we show the existence of normalized solutions by verifying the \emph{(PS)} condition at the corresponding mountain pass level for any $\mu>0$. The contribution of this paper is that the recent results obtained for $L^2$-subcritical perturbation by Chen et al. (J. Geom. Anal. 33, 371 (2023)) is extended to the case $\bar{p}\leq p<4^*$. Moreover, we also discuss asymptotic behavior of the energy to the mountain pass solution when $\mu\to 0^+$ and $c\to 0^+$, respectively.