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

TL;DR

This paper investigates the existence of solutions to a singularly perturbed energy critical Choquard equation in \\mathbb{R}^N, demonstrating how solutions can be constructed for small perturbations using Lyapunov-Schmidt reduction.

Contribution

The paper introduces a novel application of Lyapunov-Schmidt reduction to construct solutions for the perturbed energy critical Choquard equation in \\mathbb{R}^N.

Findings

Solutions of the form u_{\\eps} approximate unperturbed solutions as epsilon approaches zero.

Construction of solutions is valid for sufficiently small epsilon.

The solutions resemble scaled and translated positive solutions of the unperturbed equation.

Abstract

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*}