Uniqueness of solution for logarithm Choquard equation

Abstract

In this paper, we prove the uniqueness of positive solutions for the following Choquard equation involving logarithm convolution

\begin{equation*}-\Delta u(x)=e^{[\int_{\mathbb R^N} \ln{\frac {|y|}{|x-y|}}u(y)^{\frac{2N}{N-2}}\,dy]}u(x)^{\frac{N}{N-2}}\quad {\rm in}\ \mathbb{R}^N\end{equation*}

where $N\geq 3$. Under the assumptions that

\begin{equation*}{\int_{\mathbb R^N}} e^{\frac{N+2}2[\int_{\mathbb R^N} \ln{\frac {|y|}{|x-y|}}u(y)^{\frac{2N}{N-2}}\,dy]}\,dx \lt \infty,\ {\int_{\mathbb R^N}} u^{\frac{N+2}{N-2}}\,dx \lt \infty \quad{\rm and }\quad {\int_{\mathbb R^N}} u^{\frac{2N}{N-2}}\,dx \lt \infty,\end{equation*}

we show that any positive solution of the above equation must have the following form

\begin{equation*}u(x)=(\frac{C\varepsilon}{\varepsilon^2+|x-x_0|^2})^{\frac{N-2}2},\end{equation*}

where $C$ is a positive constant, $\varepsilon \gt 0$ and $x_0\in \mathbb R^N$ are two parameters.