Nondegeneracy of positive solutions for a biharmonic Hartree equation and its application

Abstract

We study the nondegeneracy of positive solutions of the following biharmonic Hartree equation${\Delta }^{2}u=\left(\right|x{|}^{-\alpha }⁎\left|u{|}^p\right)u^{p-1},\text{in}{R}^N,$ where $p=\frac{2N-\alpha }{N-4}$, $N\ge 5$ and $0<\alpha <N$. Our method relies on the spherical harmonic decomposition and the Funk-Heck formula of the spherical harmonic functions. Then as an application, by applying a finite dimension reduction and local Pohožaev identity, we can construct multi-bubble solutions for the following equation with potential${\Delta }^{2}u+V\left(\right|x^{\prime }|,x^{″})u=\left(\right|x{|}^{-\alpha }⁎\left|u{|}^p\right)u^{p-1}x\in R^N,$ where $N\ge 9$, $(x^{\prime },x^{″})\in R^2\times R^{N-2}$ and $V\left(\right|x^{\prime }|,x^{″})$ is a bounded and nonnegative function. We prove that the existence result is restricted to the range $6-\frac{12}{N-4}\le \alpha <N$ which shows the influence of the order of Riesz potential.