Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two

Abstract

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\left(\int\limits_{\Omega}\frac{u^{p+1}(y)}{|x-y|^\alpha}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ \Omega, \quad \ \ u=0, \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $\alpha\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $\Omega$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $\alpha \to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation.