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

TL;DR

This paper proves the existence of normalized solutions for a focusing biharmonic Hartree equation with a local perturbation, extending previous results to a broader parameter range and analyzing the solutions' asymptotic behavior.

Contribution

It extends recent existence results for $L^2$-subcritical perturbations to the case $ar{p}\leq p<4^*$ and investigates the asymptotic behavior of solutions as parameters approach zero.

Findings

Existence of normalized solutions at mountain pass level.

Extension of previous results to a larger parameter range.

Analysis of energy behavior as perturbation and mass tend to zero.

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.