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.
Categories: math.AP