Nondegeneracy of positive solutions for a biharmonic hartree equation and its applications

Abstract

In this paper, we are interested in some problems related to the following biharmonic hartree equation \begin{equation*} \Delta^{2} u=(|x|^{-\alpha}\ast |u|^{p})u^{p-1},\sp \text{in}\quad\R^N. \end{equation*} where $p=\frac{2N-\alpha}{N-4}$, $N\geq 9$ and $0<\alpha<N$. First, by using the spherical harmonic decomposition and the Funk-Heck formula of the spherical harmonic functions, we prove the nondegeneracy of the positive solutions of the above biharmonic equation. As applications, we investigate the stability of a version of nonlocal Sobolev inequality \begin{equation} \int_{\R^N}|\Delta u|^{2} \geq S^{*}\left( \int_{\R^N}\Big(|x|^{-\alpha}\ast |u|^{p}\Big)u^{p} dx\right)^{\frac{1}{p}}, \end{equation} and give a gradient form remainder. Moreover, by applying a finite dimension reduction and local Poho$\check{z}$aev identity, we can also construct multi-bubble solutions for the following equation with potential \begin{equation*} \Delta^2 u+V(|x'|, x'')u =\Big(|x|^{-\alpha}\ast |u|^{p}\Big)u^{p-1}\hspace{4.14mm}x\in \mathbb{R}^N. \end{equation*} where $N\geq9$, $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$ and $V(|x'|, x'')$ is a bounded and nonnegative function. We will show what is the role fo the dimension and the order of the Riesz potential in proving the existence result. In fact, the existence result is restricted to the range $6-\frac{12}{N-4}\leq\alpha<N$.