On positive solutions of critical semilinear equations involving the Logarithmic Laplacian

Abstract

In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E){\mathcal{L}}_{\Delta }u=ku\log u,u\ge 0\mathrm{in}{\mathbb{R}}^n,$$ 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 $k=\frac{4}{n}$, problem $(E)$ only has the solutions with the form $$u_{\tilde{x},t}(x)={\beta }_n(\frac{t}{t^2+|x-\tilde{x}{|}^2)}{)}^{\frac{n}{2}}\text{ for any t>0, x~∈Rn},$$ where $n\in \mathbb{N}$, ${\beta }_n=2^{\frac{n}{2}}{e}^{\frac{n}{2}\psi (\frac{n}{2})}>0$. When $k\in (0,+\infty )\setminus \{\frac{4}{n}\}$, problem $(E)$ has no any positive solution.

35 pages