A perturbation result for the energy critical Choquard equation in $\mathbb{R}^N$

We study the singularly perturbed nonlinear energy critical Choquard equation \begin{equation*} -{\Laplace u}\qty({x}) -{\alpha} \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^{\lambda}}\odif{y} u^{p-1}\qty({x}) -\eps k\qty(x)u^{\frac{N+2}{N-2}}\qty(x)=0, \qquad x\in\R^N, \end{equation*} where $N\geq 3$, $0<\lambda<N$, $\lambda\leq 4$, $p=\frac{2N-\lambda}{N-2}$, $\alpha = \frac{ N\qty({N-2})\fct{\Gamma}{N-\frac{\lambda}{2}} }{ \pi^{\frac{N}{2}}\fct{\Gamma}{\frac{N-\lambda}{2}} }$,~ and $k$ is a positive function. By making use of a Lyapunov-Schmidt reduction argument, for sufficiently small $\eps>0$, we construct solutions of the form \begin{align*} u_{\eps}\qty(x)=U_{\mu_{\eps},\xi_{\eps}}\qty(x)\qty(1+\O\qty(\eps)), \end{align*} where $U_{\mu_{\eps},\xi_{\eps}}$ is a positive solution of the unperturbed equation \begin{equation*} -{\Laplace u}\qty({x}) -{\alpha} \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^{\lambda}}\odif{y}=0,\qquad x\in\R^N. \end{equation*}