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 $$(E){\mathcal{L}}_{\mathrm{\Delta }}u=ku\log u,u\ge 0\mathrm{i}\mathrm{n}{\mathbb{R}}^n,$$ where $k\in (0,+\mathrm{\infty })$, ${\mathcal{L}}_{\mathrm{\Delta }}$ is the logarithmic Laplacian in ${\mathbb{R}}^n$ with $n\in \mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac{4}{n}$, problem $(E)$ only has the solutions with the form $$u_{\tilde{x},t}(x)={\beta }_n(\frac{t}{t^2+\mathrm{\mid }x-\tilde{x}{\mathrm{\mid }}^2)}{)}^{\frac{n}{2}}\text{ for any }t>0\text{, }\tilde{x}\in {\mathbb{R}}^n,$$ where $n\in \mathbb{N}$, ${\beta }_n=2^{\frac{n}{2}}{e}^{\frac{n}{2}\psi (\frac{n}{2})}>0$. When $k\in (0,+\mathrm{\infty })\setminus \{\frac{4}{n}\}$, problem $(E)$ has no any positive solution.