On positive solutions of critical semilinear equations involving the Logarithmic Laplacian

In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_\Delta u= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n, \qquad\qquad\qquad\qquad\qquad\qquad$$ where $k\in(0,+\infty)$, $\mathcal{L}_\Delta$ is the logarithmic Laplacian in $\mathbb{R}^n$ with $n\in\mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\tilde x,t}(x)=\beta_n \Big(\frac{t}{t^2+|x-\tilde x|^2)}\Big)^{\frac{n}{2}}\quad \text{ for any $t>0$, $\tilde x\in\mathbb{R}^n$},$$ where $n\in\mathbb{N}$, $\beta_n=2^{\frac n2} e^{\frac n2\psi(\frac n2) }>0$. When $k\in(0,+\infty)\setminus\{\frac 4n\}$, problem $(E)$ has no any positive solution.