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

Abstract

We study the singularly perturbed nonlinear energy critical Choquard equation \begin{equation*} -{\Laplace u}\qty({x}) -α \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^λ}\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 $N\ge 3$, $0<\lambda <N$, $\lambda \le 4$, $p=\frac{2N-\lambda }{N-2}$, $α= \frac{ N\qty({N-2})\fctΓ{N-\fracλ{2}} }{ π^{\frac{N}{2}}\fctΓ{\frac{N-λ}{2}} }$,~ 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_{μ_{\eps},ξ_{\eps}}\qty(x)\qty(1+Ø\qty(\eps)), \end{align*} where $U_{μ_{\eps},ξ_{\eps}}$ is a positive solution of the unperturbed equation \begin{equation*} -{\Laplace u}\qty({x}) -α \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^λ}\odif{y}=0,\qquad x\in\R^N. \end{equation*}

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