A Liouville-type theorem in conformally invariant equations

Abstract

Given a smooth function $K(x)$ satisfying a polynomially cone condition and $x\cdot \nabla K\le 0$, we prove that there is no solution $u\in C^{\infty }({\mathbb{R}}^2)$ of the equation $$-\Delta u=K(x)e^{2u}\mathrm{on}{\mathbb{R}}^2$$ with $u\le C$ and ${\int }_{{\mathbb{R}}^2}|K(x)|e^{2u}dx<+\infty$. As a consequence, there is no such solution if $K(x)$ is a non-constant polynomial with $x\cdot \nabla K\le 0$. The latter result already includes a result of Struwe(JEMS 2020) as a particular case. Higher order cases are set up with additional assumption on the behavior of $\Delta u$ near infinity.