Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions

We study nondegeneracy of ground states of the Hartree equation $$-\mathrm{\Delta }u+u=(I_2\ast u^2)uin{\mathbb{R}}^n$$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov-Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$-{\epsilon }^2\mathrm{\Delta }u+u+V(x)u={\epsilon }^{-2}(I_2\ast u^2)uin{\mathbb{R}}^n\mathrm{.}$$